Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://wasp.phys.msu.ru/forum/index.php?act=attach&type=post&id=12135 Äàòà èçìåíåíèÿ: Unknown Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 07:10:20 2012 Êîäèðîâêà:

arXiv:1111.6521v1 [math.HO] 28 Nov 201

MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION BASHKIR STATE UNIVERSITY

RUSLAN A. SHARIPOV

COURSE OF ANALYTICAL GEOMETRY

The textbook

UFA 2011


2

UDK 514.123 BBK 22.151 X25 Referee: The division of Mathematical Analysis of Bashkir State Pedagogical University (BGPU) named after Miftakhetdin Akmulla, Ufa. used the computer ackage and I used the CyrTUG assoof this b ook is also

In preparing Russian edition of this b ook I typ esetting on the base of the AMS-TEX p Cyrillic fonts of the Lh-family distributed by ciation of Cyrillic TEX users. English edition typ eset by means of the AMS-TEX package.
X25

X aripov R. A. Kurs analitiqesko geometrii: Uqebnoe posobie / R. A. Xaripov. -- Ufa: RIC BaxGU, 2010. -- 228 s.

ISBN 978-5-7477-2574-4
Uqebnoe posobie po kursu analitiqesko geometrii adresovano studentam matematikam, fizikam, a take studentam ienerno-tehniqeskih, tehnologiqeskih i inyh specialnoste , dl kotoryh gosudarstvennye obrazovatelnye standarty predusmatriva t izuqenie dannogo predmeta. UDK 514.123 BBK 22.151

ISBN 978-5-7477-2574-4 English Translation

c Sharip ov R.A., 2010 c Sharip ov R.A., 2011


CONTENTS. CONTENTS. ... .... ... .... .... ... .... .... ... .... .... ... .... .... ... .... .... ... .... . 3. PREFACE. .. .... .... ... .... .... ... .... .... ... .... .... ... .... .... ... .... .... ... .... . 7. CHAPTER I. VECTOR ALGEBRA. . .... ... .... .... ... .... .... ... .... . 9. § 1. Three-dimensional Euclidean space. Acsiomatics and visual evidence. ... .... .... ... .... .... ... .... .... ... .... ... § 2. Geometric vectors. Vectors b ound to p oints. . .... ... § 3. Equality of vectors. .. ... .... .... ... .... .... ... .... .... ... .... ... § 4. The concept of a free vector. .. .... .... ... .... .... ... .... ... § 5. Vector addition. ... .... ... .... .... ... .... .... ... .... .... ... .... ... § 6. Multiplication of a vector by a numb er. . .... ... .... ... § 7. Prop erties of the algebraic op erations with vectors. § 8. Vectorial expressions and their transformations. ... § 9. Linear combinations. Triviality, non-triviality, and vanishing. . .... .... ... .... .... ... .... .... ... .... .... ... .... ... § 10. Linear dep endence and linear indep endence. . .... ... § 11. Prop erties of the linear dep endence. .. .... .... ... .... ... § 12. Linear dep endence for n = 1. . .... .... ... .... .... ... .... ... § 13. Linear dep endence for n = 2. Collinearity of vectors. .... ... .... .... ... .... .... ... .... .... ... .... .... ... .... ... § 14. Linear dep endence for n = 3. Coplanartity of vectors. .... ... .... .... ... .... .... ... .... .... ... .... .... ... .... ... § 15. Linear dep endence for n 4. . .... .... ... .... .... ... .... ... § 16. Bases on a line. ... .... ... .... .... ... .... .... ... .... .... ... .... ... § 17. Bases on a plane. .... ... .... .... ... .... .... ... .... .... ... .... ... § 18. Bases in the space. .. ... .... .... ... .... .... ... .... .... ... .... ... § 19. Uniqueness of the expansion of a vector in a basis. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. 11. 13. 14. 16. 18. 21. 28. 32. 34. 36. 37.

. . . .

. ... ... 38. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40. 42. 45. 46. 48. 50.


4

CONTENTS.

§ 20. Index setting convention. .. .... ... .... .... ... .... .... ... .... . § 21. Usage of the coordinates of vectors. .. ... .... .... ... .... . § 22. Changing a basis. Transition formulas and transition matrices. .... .... ... .... .... ... .... .... ... .... . § 23. Some information on transition matrices. . .... ... .... . § 24. Index setting in sums. .. .... .... ... .... .... ... .... .... ... .... . § 25. Transformation of the coordinates of vectors under a change of a basis. . .... ... .... .... ... .... .... ... .... . § 26. Scalar product. . .... .... ... .... .... ... .... .... ... .... .... ... .... . § 27. Orthogonal pro jection onto a line. .... ... .... .... ... .... . § 28. Prop erties of the scalar product. .. .... ... .... .... ... .... . § 29. Calculation of the scalar product through the coordinates of vectors in a skew-angular basis. § 30. Symmetry of the Gram matrix. .... .... ... .... .... ... .... . § 31. Orthonormal basis. .... ... .... .... ... .... .... ... .... .... ... .... . § 32. The Gram matrix of an orthonormal basis. .. ... .... . § 33. Calculation of the scalar product through the coordinates of vectors in an orthonormal basis. § 34. Right and left triples of vectors. The concept of orientation. . ... .... .... ... .... .... ... .... . § 35. Vector product. . .... .... ... .... .... ... .... .... ... .... .... ... .... . § 36. Orthogonal pro jection onto a plane. . ... .... .... ... .... . § 37. Rotation ab out an axis. .... .... ... .... .... ... .... .... ... .... . § 38. The relation of the vector product with pro jections and rotations. .... .... ... .... .... ... .... . § 39. Prop erties of the vector product. .. .... ... .... .... ... .... . § 40. Structural constants of the vector product. .. ... .... . § 41. Calculation of the vector product through the coordinates of vectors in a skew-angular basis. § 42. Structural constants of the vector product in an orthonormal basis. ... .... ... .... .... ... .... .... ... .... . § 43. Levi-Civita symb ol. ... ... .... .... ... .... .... ... .... .... ... .... . § 44. Calculation of the vector product through the coordinates of vectors in an orthonormal basis. .. .... . § 45. Mixed product. . .... .... ... .... .... ... .... .... ... .... .... ... .... .

... ... . 51. ... ... . 52. ... ... . 53. ... ... . 57. ... ... . 59. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63. 65. 67. 73. 75. 79. 80. 81.

. .. .. ..

. ... . 82. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83. 84. 86. 88.

... ... . 91. ... ... . 92. ... ... . 95. .. ... . 96. ... ... . 97. ... ... . 99. ... .. 102. ... .. 104.


CONTENTS.

5

§ 46. Calculation of the mixed product through the coordinates of vectors in an orthonormal basis. .... .... § 47. Prop erties of the mixed product. .... ... .... .... ... .... .... § 48. The concept of the oriented volume. .. .... .... ... .... .... § 49. Structural constants of the mixed product. ... .... .... § 50. Calculation of the mixed product through the coordinates of vectors in a skew-angular basis. . .... .... § 51. The relation of structural constants of the vectorial and mixed products. ... .... .... ... .... .... ... .... .... ... .... .... § 52. Effectivization of the formulas for calculating vectorial and mixed products. .... .... ... .... .... ... .... .... § 53. Orientation of the space. .... ... .... .... ... .... .... ... .... .... § 54. Contraction formulas. . .... .... ... .... .... ... .... .... ... .... .... § 55. The triple product expansion formula and the Jacobi identity. .. ... .... .... ... .... .... ... .... .... ... .... .... § 56. The product of two mixed products. .. .... .... ... .... ....

. . . .

. . . .

. . . .

. . . .

105. 108. 111. 113.

... . 115. ... . 116. ... . 121. ... . 124. ... . 125. ... . 131. ... . 134.

CHAPTER I I. GEOMETRY OF LINES AND SURFACES. .. ... .... .... ... .... .... ... .... .... ... .... .... ... . 139. § § § § § § § 1. 2. 3. 4. 5. 6. 7. Cartesian coordinate systems. . .... .... ... .... .... ... .. Equations of lines and surfaces. .. .... ... .... .... ... .. A straight line on a plane. ... ... .... .... ... .... .... ... .. A plane in the space. .. .... .... ... .... .... ... .... .... ... .. A straight line in the space. . ... .... .... ... .... .... ... .. Ellipse. Canonical equation of an ellipse. .... ... .. The eccentricity and directrices of an ellipse. The prop erty of directrices. . ... .... .... ... .... .... ... .. The equation of a tangent line to an ellipse. .. .. Focal prop erty of an ellipse. ... .... .... ... .... .... ... .. Hyp erb ola. Canonical equation of a hyp erb ola. The eccentricity and directrices of a hyp erb ola. The prop erty of directrices. . ... .... .... ... .... .... ... .. The equation of a tangent line to a hyp erb ola. Focal prop erty of a hyp erb ola. ... .... ... .... .... ... .. Asymptotes of a hyp erb ola. ... .... .... ... .... .... ... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139. 141. 142. 148. 154. 160. 165. 167. 170. 172. 179. 181. 184. 186.

§ 8. § 9. § 10. § 11. § 12. § 13. § 14.

.. .. .. .. .. .. . .. . . . . . . . . . . . . . . . .


6

CONTENTS.

§ § § § § § §

15. 16. 17. 18. 19. 20. 21.

§ 22. § § § § 23. 24. 25. 26.

Parab ola. Canonical equation of a parab ola. The eccentricity of a parab ola. . .... .... ... .... .... The equation of a tangent line to a parab ola. Focal prop erty of a parab ola. ... .... .... ... .... .... The scale of eccentricities. .... ... .... .... ... .... .... Changing a coordinate system. . .... .... ... .... .... Transformation of the coordinates of a p oint under a change of a coordinate system. .... .... Rotation of a rectangular coordinate system on a plane. The rotation matrix. .. .... ... .... .... Curves of the second order. ... ... .... .... ... .... .... Classification of curves of the second order. . Surfaces of the second order. . ... .... .... ... .... .... Classification of surfaces of the second order.

... ... .. ... ... ...

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

187. 190. 190. 193. 194. 195.

... .... .... .. 196. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197. 199. 200. 206. 207.

REFERENCES. ... ... .... .... ... .... .... ... .... .... ... .... .... ... .... .... .. 216. CONTACT INFORMATION. .. .... ... .... .... ... .... .... ... .... .... .. 217. APPENDIX. .... .... ... .... .... ... .... .... ... .... .... ... .... .... ... .... .... .. 218.


PREFACE.

The elementary geometry, which is learned in school, deals with basic concepts such as a point, a straight line, a segment. They are used to comp ose more complicated concepts: a polygonal line, a polygon, a polyhedron. Some curvilinear forms are also considered: a circle, a cylinder, a cone, a sphere, a bal l. The analytical geometry basically deals with the same geometric ob jects as the elementary geometry does. The difference is in a method for studying these ob jects. The elementary geometry relies on visual impressions and formulate the prop erties of geometric ob jects in its axioms. From these axioms various theorems are derived, whose statements in most cases are also revealed in visual impressions. The analytical geometry is more inclined to a numeric description of geometric ob jects and their prop erties. The transition from a geometric description to a numeric description b ecomes p ossible due to coordinate systems. Each coordinate system associate some groups of numb ers with geometric p oints, these groups of numb ers are called coordinates of p oints. The idea of using coordinates in geometry b elongs French mathematician Rene Descartes. Simplest coordinate systems suggested by him at present time are called Cartesian coordinate systems. The construction of Cartesian coordinates particularly and the analytical geometry in general are based on the concept of a vector. The branch of analytical geometry studying vectors is called the vector algebra. The vector algebra constitutes the first chapter of this b ook. The second chapter explains the theory of straight lines and planes and the theory of curves of the second order. In the third chapter the theory of surfaces of the second order is explained in brief. The b ook is based on lectures given by the author during several years in Bashkir State University. It was planned as the first b ook in a series of three b ooks. However, it happ ens that
CopyRight c Sharipov R.A., 2010.


8

PREFACE.

the second published b ­ «Course ­ «Course

and the third efore the first of linear algeb of differential

b ooks in this series were written and b ook. These are ra and multidimensional geometry» [1]; geometry» [2].

Along with the ab ove b ooks, the following b ooks were written: ­ «Representations of finite group» [3]; ­ «Classical electrodynamics and theory of relativity» [4]; ­ «Quick introduction to tensor analysis» [5]. ­ «Foundations of geometry for university students and high school students» [6]. The b ook [3] can b e considered as a continuation of the b ook [1] which illustrates the application of linear algebra to another branch of mathematics, namely to the theory of groups. The b ook [4] can b e considered as a continuation of the b ook [2]. It illustrates the application of differential geometry to physics. The b ook [5] is a brief version of the b ook [2]. As for the b ook [6], by its sub ject it should precede this b ook. It could br recommended to the reader for deep er logical understanding of the elementary geometry. I am grateful to Prof. R. R. Gadylshin and Prof. D. I. Borisov for reading and refereeing the manuscript of this b ook and for valuable advices.

Decemb er, 2010.

R. A. Sharip ov.


CHAP TE R I

VECTOR ALGEBRA.

§ 1. Three-dimensional Euclidean space. Acsiomatics and visual evidence. Like the elementary geometry explained in the b ook [6], the analytical geometry in this b ook is a geometry of threedimensional space E. We use the symb ol E for to denote the space that we observe in our everyday life. Despite b eing seemingly simple, even the empty space E p ossesses a rich variety of prop erties. These prop erties reveal through the prop erties of various geometric forms which are comprised in this space or p otentially can b e comprised in it. Systematic study of the geometric forms in the space E was initiated by ancient Greek mathematicians. It was Euclid who succeeded the most in this. He has formulated the basic properties of the space E in five p ostulates, from which he derived all other prop erties of E. At the present time his p ostulates are called axioms. On the basis of modern requirements to the rigor of mathematical reasoning the list of Euclid's axioms was enlarged from five to twenty. These twenty axioms can b e found in the b ook [6]. In favor of Euclid the space that we observe in our everyday life is denoted by the symb ol E and is called the three-dimensional Euclidean space. The three-dimensional Euclidean p oint space E consists of p oints. All geometric forms in it also consist of p oints. subsets of the space E. Among subsets of the space E straight lines


10

CHAPTER I. VECTOR ALGEBRA.

and planes (see Fig. 1.2) play an esp ecial role. They are used in the statements of the first eleven Euclid's axioms. On the base of these axioms the concept of a segment (see Fig. 1.2) is introduced. The concept of a segment is used in the statement of the twelfth axiom. The first twelve of Euclid's axioms app ear to b e sufficient to define the concept of a ray and the concept of an angle b etween two rays outgoing from the same p oint. The concepts

of a segment and an angle along with the concepts of a straight line and a plane app ear to b e sufficient in order to formulate the remaining eight Euclid's axioms and to build the elementary geometry in whole. Even the ab ove survey of the b ook [6], which is very short, shows that building the elementary geometry in an axiomatic way on the basis of Euclid's axioms is a time-consuming and lab orious work. However, the reader who is familiar with the elementary geometry from his school curriculum easily notes that proof of theorems in school textb ooks are more simple than those in [6]. The matter is that proofs in school textb ooks are not proofs in the strict mathematical sense. Analyzing them carefully, one can find in these proofs the usage of some non-proved prop ositions which are visually obvious from a supplied drawing since we have a rich exp erience of living within the space E. Such proofs can b e transformed to strict mathematical proofs by filling omissions,


§ 2. GEOMETRIC VECTORS.

11

i. e. by proving visually obvious prop ositions used in them. Unlike [6], in this b ook I do not load the reader by absolutely strict proofs of geometric results. For geometric definitions, constructions, and theorems the strictness level is used which is close to that of school textb ooks and which admits drawings and visually obvious facts as arguments. Whenever it is p ossible I refer the reader to strict statements and proofs in [6]. As far as the analytical content is concerned, i. e. in equations, in formulas, and in calculations the strictness level is applied which is habitual in mathematics without any deviations. § 2. Geometric vectors. Vectors b ound to p oints. - - Definition 2.1. A geometric vectors AB is a straight line segment in which the direction from the p oint A to the p oint B is sp ecified. The p oint A is called the initial point of the vector - - AB , while the p oint B is called its terminal point. - - The direction of the vector AB in drawing is marked by an arrow (see Fig. 2.1). For this reason vectors sometimes are called directed segments. Each segment [AB ] is associated with two different vectors: - - - - - - AB and B A . The vector B A is usually called the opposite vector - - for the vector AB . Note that the arrow sign on - - the vector AB and b old dots at the ends of the segment [AB ] are merely symb olic signs used to make the drawing more clear. - - When considered as sets of p oints the vector AB and the segment [AB ] do coincide. A direction on a segment, which makes it a vector, can mean different things in different situations. For instance, drawing a - - vector AB on a geographic map, we usually mark the displacement of some ob ject from the p oint A to the p oint B . However,


12

CHAPTER I. VECTOR ALGEBRA.

- - if it is a weather map, the same vector AB can mean the wind direction and its sp eed at the p oint A. In the first case the - - length of the vector AB is prop ortional to the distance b etween - - the p oints A and B . In the second case the length of AB is prop ortional to the wind sp eed at the p oint A. There is one more difference in the ab ove two examples. In - - the first case the vector AB is b ound to the p oints A and B by - - its meaning. In the second case the vector AB is b ound to the p oint A only. The fact that its arrowhead end is at the p oint B is a pure coincidence dep ending on the scale we used for translating the wind sp eed into the length units on the map. According to what was said, geometric vectors are sub divided into two typ es: 1) purely geometric; 2) conditionally geometric. Only displacement vectors b elong to the first typ e; they actually bind some two p oints of the space E. The lengths of these vectors are always measured in length units: centimeters, meters, inches, feets etc. Vectors of the second typ e are more various. These are velocity vectors, acceleration vectors, and force vectors in mechanics; intensity vectors for electric and magnetic fields, magnetization vectors in magnetic materials and media, temp erature gradients in non-homogeneously heated ob jects et al. Vectors of the second typ e have a geometric direction and they are b ound to some p oint of the space E, but they have not a geometric length. Their lengths can b e translated to geometric lengths only up on choosing some scaling factor. Zero vectors or null vectors hold a sp ecial p osition among geometric vectors. They are defined as follows. Definition 2.2. A geometric vector of the space E whose initial and terminal p oints do coincide with each other is called a zero vector or a nul l vector. A geometric null vector can b e either a purely geometric vector


§ 3. EQUALITY OF VECTORS.

13

or a conditionally geometric vector dep ending on its nature. § 3. Equality of vectors. - - - - Definition 3.1. Two geometric vectors AB and C D are called equal if they are equal in length and if they are codirected, - - - - i. e. |AB | = |C D | and AB C D . - - - - The vectors AB and C D are said to b e codirected if they lie on a same line as shown in Fig. 3.1 of if they lie on parallel lines as shown in Fig. 3.2. In b oth cases they should b e p ointing in the

same direction. Codirectedness of geometric vectors and their equality are that very visually obvious prop erties which require substantial efforts in order to derive them from Euclid's axioms (see [6]). Here I urge the reader not to focus on the lack of rigor in statements, but b elieve his own geometric intuition. Zero geometric vectors constitute a sp ecial case since they do not fix any direction in the space. Definition 3.2. All null vectors are assumed to b e codirected to each other and each null vector is assumed to b e codirected to any nonzero vector. The length of all null vectors is zero. However, dep ending on the physical nature of a vector, its zero length is complemented with a measure unit. In the case of zero force it is zero newtons,


14

CHAPTER I. VECTOR ALGEBRA.

in the case of zero velocity it is zero meters p er second. For this reason, testing the equality of any two zero vectors, one should take into account their physical nature. Definition 3.3. All null vectors of the same physical nature are assumed to b e equal to each other and any nonzero vector is assumed to b e not equal to any null vector. Testing the equality of nonzero vectors by means of the definition 3.1, one should take into account its physical nature. The equality |AB | = |C D | in this definition assumes not only the - - - - equality of numeric values of the lengths of AB and C D , but assumes the coincidence of their measure units as well. A remark. Vectors are geometric forms, i. e. they are sets of p oints in the space E. However, the equality of two vectors introduced in the definition 3.1 differs from the equality of sets. § 4. The concept of a free vector. Defining the equality of vectors, it is convenient to use parallel translations. Each parallel translation is a sp ecial transformation of the space p : E E under which any straight line is mapp ed onto itself or onto a parallel line and any plane is mapp ed onto itself or onto a parallel plane. When applied to vectors, parallel translation preserve their length and their direction, i. e. they map each vector onto a vector equal to it, but usually b eing in a different place in the space. The numb er of parallel translations is infinite. As app ears, the parallel translations are so numerous that they can b e used for testing the equality of vectors. - - Definition 4.1. A geometric vector C D is called equal to a - - geometric vector AB if there is a parallel translation p : E E - - - - that maps the vector AB onto the vector C D , i. e. such that p(A) = C and p(B ) = D . The definition 4.1 is equivalent to the definition 3.1. I do not prove this equivalence, relying on its visual evidence and
CopyRight c Sharipov R.A., 2010.


§ 4. THE CONCEPT OF A FREE VECTOR.

15

assuming the reader to b e familiar with parallel translations from the school curriculum. A more meticulous reader can see the theorems 8.4 and 9.1 in Chapter VI of the b ook [6]. Theorem 4.1. For any two p oints A and C in the space E there is exactly one parallel translation p : E E mapping the p oint A onto the p oint C , i. e. such that p(A) = C . The theorem 4.1 is a visually obvious fact. On the other hand it coincides with the theorem 9.3 from Chapter VI in the b ook [6], where it is proved. For these two reasons we exploit the theorem 4.1 without proving it in this b ook. - - Lei's apply the theorem 4.1 to some geometric vector AB . Let C b e an arbitrary p oint of the space E and let p b e a parallel translation taking the p oint A to the p oint C . The existence and uniqueness of such a parallel translation are asserted by the theorem 4.1. Let's define the p oint D by means of the formula D = p(B ). Then, according to the definition 4.1, we have - - - - AB = C D . - - These considerations show that each geometric vector AB has a copy equal to it and attached to an arbitrary p oint C E. In the other words, by means of parallel translations each geometric - - vector AB can b e replicated up to an infinite set of vectors equal to each other and attached to all p oints of the space E. Definition 4.2. A free vector is an infinite collection of geometric vectors which are equal to each other and whose initial p oints are at all p oints of the space E. Each geometric vector in this infinite collection is called a geometric realization of a given free vector. Free vectors can b e comp osed of purely geometric vectors or of conditionally geometric vectors as well. For this reason one can consider free vectors of various physical nature.


16

CHAPTER I. VECTOR ALGEBRA.

In drawings free vectors are usually presented by a single geometric realization or by several geometric realizations if needed. Geometric vectors are usually denoted by two capital letters: - - - - - - AB , C D , E F etc. Free vectors are denoted by single lowercase letters: a, b, c etc. Arrows over these letters are often omitted since it is usually clear from the context that vectors are considered. Below in this b ook I will not use arrows in denoting free vectors. However, I will use b oldface letters for them. In many other b ooks, but not in my b ook [1], this restriction is also removed. § 5. Vector addition. Assume that two free vectors a and b are given. Let's choose some arbitrary p oint A and consider the geometric realization of the vector a with the initial p oint A. Let's denote through B the terminal p oint of this geometric realization. As a result we - - get a = AB . Then we consider the geometric realization of the vector b with initial p oint B and denote through C its terminal - - p oint. This yields b = B C . - - Definition 5.1. The geometric vector AC connecting the - - initial p oint of the vector AB with the terminal p oint of the - - - - - - vector B C is called the sum of the vectors AB and B C : - - - - - - AC = AB + B C . (5.1) - - - - The vector AC constructed by means of the vectors a = AB - - and b = B C can b e replicated up to a free vector c by parallel translations to all p oints of the space E. Such a vector c is naturally called the sum of the free vectors a and b. For this vector we write c = a + b. The correctness of such a definition is guaranteed by the following lemma. - - Lemma 5.1. The sum c = a + b = AC of two free vectors - - - - a = AB and b = B C expressed by the formula (5.1) does not


§ 5. VECTOR ADDITION.

17

dep end on the choice of a p oint A at which the geometric realiza- - tion AB of the vector a b egins. Proof. In addition to A, let's choose another initial p oint E . Then in the ab ove construction of the sum a + b the vector a has

- - two geometric realizations AB and - - two geometric realizations B C and - - - - AB = E F ,

- - E F . The vector b also has - - F G (see Fig. 5.1). Then - - - - BC = F G . (5.2)

Instead of (5.1) now we have two equalities - - - - - - AC = AB + B C , - - - - - - EG = EF + F G . (5.3)

Let's denote through p a parallel translation that maps the p oint A to the p oint E , i. e. such that p(A) = E . Due to the theorem 4.1 such a parallel translation does exist and it is unique. From p(A) = E and from the first equality (5.2), applying the definition 4.1, we derive p(B ) = F . Then from p(B ) = F and from the second equality (5.2), again applying the definition 4.1, we get p(C ) = G. As a result we have p(A) = E , p(C ) = G. (5.4)


18

CHAPTER I. VECTOR ALGEBRA.

The relationships - - the vector AC to - - fact yields AC =

(5.4) mean the vector - - E G . Now f

that the parallel translation p maps - - E G . Due to the definition 4.1 this rom the equalities (5.3) we derive (5.5)

- - - - - - - - AB + B C = E F + F G . The equalities (5.5) complete the proof of the lemma 5.1.

The addition rule given by the formula (5.1) is called the triangle rule. It is associated with the triangle AB C in Fig. 5.1. § 6. Multiplication of a vector by a numb er. Let a b e some free vector. Let's choose some arbitrary p oint A and consider the geometric realization of the vector a with initial p oint A. Then we denote through B the terminal p oint of this geometric realization of a. Let b e some numb er. It can b e

either p ositive, negative, or zero. Let > 0. In this case we lay a p oint C onto the line AB so that the following conditions are fulfilled: - - - - AC AB , |AC | = || · |AB |. (6.1)

As a result we obtain the drawing which is shown in Fig. 6.1. If = 0, we lay the p oint C so that it coincides with the p oint - - A. In this case the vector AC app ears to b e zero as shown in Fig. 6.2 and we have the relationship |AC | = || · |AB |. (6.2)


§ 6. MULTIPLICATION OF A VECTOR BY A NUMBER.

19

In the case < 0 we lay the p oint C onto the line AB so that the following two conditions are fulfilled: - - - - AC AB , |AC | = || · |AB |. (6.3)

This arrangement of p oints is shown in Fig. 6.3. Definition 6.1. In each of the three cases > 0, = 0, and - - - - < 0 the geometric vector AC defined through the vector AB according to the drawings in Fig. 6.1, in Fig. 6.2, and in Fig. 6.3 and according to the formulas (6.1), (6.2), and (6.3) is called - - the product of the vector AB by the numb er . This fact is expressed by the following formula: - - - - AC = · AB . (6.4)

The case a = 0 is not covered by the ab ove drawings in Fig. 6.1, in Fig. 6.2, and in Fig. 6.3. In this case the p oint B coincides with the p oints A and we have |AB | = 0. In order to provide the equality |AC | = || · |AB | the p oint C is chosen coinciding with the p oint A. Therefore the product of a null vector by an arbitrary numb er is again a null vector. - - The geometric vector AC constructed with the use of the - - vector a = AB and the numb er can b e replicated up to a free vector c by means of the parallel translations to all p oints of the space E. Such a free vector c is called the product of the free vector a by the numb er . For this vector we write c = · a. The correctness of this definition of c = · a is guaranteed by the following lemma. - - Lemma 6.1. The product c = · a = AC of a free vector - - a = AB by a numb er expressed by the formula (6.4) does not dep end on the choice of a p oint A at which the geometric realization of the vector a is built.


20

CHAPTER I. VECTOR ALGEBRA.

Proof. Let's prove the lemma for the case a = 0 and > 0. In addition to A we choose another initial p oint E . Then in the

construction of the product · a the vector a gets two geometric - - - - realizations AB and E F (see Fig. 6.4). Hence we have - - - - AB = E F . (6.5)

Let's denote through p the parallel translation that maps the p oint A to the p oint E , i. e. such that p(A) = E . Then from the equality (6.5), applying the definition 4.1, we derive p(B ) = F . The p oint C is placed on the line AB at the distance |AC | = || · |AB | from the p oint A in the direction of the vector - - AB . Similarly, the p oint G is placed on the line E F at thew distance |E G| = || · |E F | from the p oint E in the direction of - - the vector E F . From the equality (6.5) we derive |AB | = |E F |. Therefore |AC | = || · |AB | and |E G| = || · |E F | mean that |AC | = |E F |. Due to p(A) = E and p(B ) = F the parallel translation p maps the line AB onto the line E F . It preserves lengths of segments and maps codirected vectors to codirected ones. Hence p(C ) = G. Along with p(A) = E due to the - - - - definition 4.1 the equality p(C ) = G yields AC = E G , i. e. - - - - · AB = · E F . The lemma 6.1 is proved for the case a = 0 and > 0. Its proof for the other cases is left to the reader as an exercise. Exercise 6.1. Consider the cases = 0 and < 0 for a = 0


§ 7. PROPERTIES OF THE ALGEBRAIC OPERATIONS . . .

21

and consider the case a = 0. Prove the lemma 6.1 for these cases and provide your proof with drawings analogous to that of Fig 6.4. § 7. Prop erties of the algebraic op erations with vectors. The addition of vectors and their multiplication by numb ers are two basic algebraic op erations with vectors in the threedimensional Euclidean p oint space E. Eight basic prop erties of these two algebraic op erations with vectors are usually considered. The first four of these eight prop erties characterize the op eration of addition. The other four characterize the op eration of multiplication of vectors by numb ers and its correlation with the op eration of vector addition. Theorem 7.1. The op eration of addition of free vectors and the op eration of their multiplication by numb ers p ossess the following prop erties: 1) commutativity of addition: a + b = b + a; 2) associativity of addition: (a + b) + c = a + (b + c); 3) the feature of the null vector: a + 0 = a; 4) the existence of an opp osite vector: for any vector a there is an opp osite vector a such that a + a = 0; 5) distributivity of multiplication over the addition of vectors: k · (a + b) = k · a + k · b; 6) distributivity of multiplication over the addition of numb ers: (k + q ) · a = k · a + q · a; 7) associativity of multiplication by numb ers: (k q ) · a = k · (q · a); 8) the feature of the numeric unity: 1 · a = a. Let's consider the prop erties listed in the theorem 7.1 one by one. Let's b egin with the commutativity of addition. The sum a + b in the left hand side of the equality a + b = b + a is calculated by means of the triangle rule up on choosing some - - - - geometric realizations a = AB and b = B C as shown in Fig. 7.1. Let's draw the line parallel to the line B C and passing through the p oint A. Then we draw the line parallel to the
CopyRight c Sharipov R.A., 2010.


22

CHAPTER I. VECTOR ALGEBRA.

line AB and passing through the p oint C . Both of these lines are in the plane of the triangle AB C . For this reason they intersect at some p oint D . The segments [AB ], [B C ], [C D ], and [D A] form a parallelogram. - - Let's mark the vectors D C - - and AD on the segments [C D ] - - and [D A]. It is easy to see that the vector D C is produced from - - the vector AB by applying the parallel translation from the p oint - - A to the p oint D . Similarly the vector AD is produced from the - - vector B C by applying the parallel translation from the p oint B - - - - - - - - to the p oint A. Therefore D C = AB = a and B C = AD = b. Now the triangles AB C and AD C yield - - - - - - AC = AB + B C = a + b, - - - - - - AC = AD + D C = b + a. (7.1)

From (7.1) we derive the required equality a + b = b + a. The relationship a + b = b + a and Fig. 7.1 yield another method for adding vectors. It is called the paral lelogram rule. In - - order to add two vectors a and b their geometric realizations AB - - and AD with the common initial p oint A are used. They are completed up to the parallelogram AB C D . Then the diagonal of this parallelogram is taken for the geometric realization of the - - - - - - sum: a + b = AB + AD = AC . Exercise 7.1. Prove the equality a + b = b + a for the case where a b. For this purp ose consider the sub cases 1) a b; 3) a b and |a| = |b|; 2) a b and |a| > |b|;

4) a b and |a| < |b|.


§ 7. PROPERTIES OF THE ALGEBRAIC OPERATIONS . . .

23

The next prop erty in the theorem 7.1 is the associativity of the op eration of vector addition. In order to prove this prop erty we choose some arbitrary initial p oint A and construct the following geometric realizations of the - - - - vectors: a = AB , b = B C , and - - c = C D . Applying the triangle rule of vector addition to the triangles AB C and AC D , we get the relationships - - - - - - a + b = AB + B C = AC , - - - - - - (a + b) + c = AC + C D = AD (7.2)

(see Fig. 7.2). Applying the same rule to the triangles B C D and AB D , we get the analogous relationships - - - - - - b + c = BC + C D = BD , - - - - - - a + (b + c) = AB + B D = AD . (7.3)

The required relationship (a + b) + c = a + (b + c) now is immediate from the formulas (7.2) and (7.3). A remark. The tetragon AB C D in Fig. 7.2 is not necessarily planar. For this reason the line C D is shown as if it goes under the line AB , while the line B D is shown going over the line AC . The feature of the null vector a + 0 = a is immediate from the triangle rule for vector addition. Indeed, if an initial p oint A for - - the vector a is chosen and if its geometric realization AB is built, then the null vector 0 is presented by its geometric realization - - - - - - - - B B . From the definition 5.1 we derive AB + B B = AB which yields a + 0 = a. The existence of an opp osite vector is also easy to prove. Assume that the vector a is presented by its geometric realization


24

CHAPTER I. VECTOR ALGEBRA.

- - - - AB . Let's consider the opp osite geometric vector B A and let's denote through a the corresp onding free vector. Then - - - - - - a + a = AB + B A = AA = 0. The distributivity of multiplication over the vector addition follows from the prop erties of the homothety transformation in the Euclidean space E (see § 11 of Chapter VI in [6]). It is sometimes called the similarity transformation, which is not quite exact. Similarity transformations constitute a larger class of transformations that comprises homothety transformations as a sub class within it. Let a b and let the sum of vectors a + b is calculated according to the triangle rule as shown in Fig. 7.3. Assume that

k > 0. Let's construct the homothety transform with the center at the p oint A and with the k. Let's denote through E the image of the p transformation hkA and let's denote through F p oint C under this transformation: E = hkA (B ),

ation hkA : E E homothety factor oint B under the the image of the

F = hkA (C ).

Due to the prop erties of the homothety the line E F is parallel to


§ 7. PROPERTIES OF THE ALGEBRAIC OPERATIONS . . .

25

the line B C and we have the following relationships: - - EF - - AE - - AF - - BC , - - AB , - - AC , |E F | = |k| · |B C |, |AE | = |k| · |AB |, |AF | = |k| · |AC |. (7.4)

Comparing (7.4) with (6.1) and taking into account that we consider the case k > 0, from (7.4) we derive - - - - AE = k · AB , - - - - EF = k · BC , - - - - AF = k · AC . (7.5)

The relationships (7.5) are sufficient for to prove the distributivity of the mutiplication of vectors by numb ers over the op eration of vector addition. Indeed, from (7.5) we obtain: - - - - - - - - k · (a + b) = k · (AB + B C ) = k · AC = AF = - - - - - - - - = AE + E F = k · AB + k · B C = k · a + k · b.

(7.6)

The case where a b and k < 0 is very similar to the case just ab ove. In this case Fig. 7.3 is replaced by Fig. 7.4. Instead of the relationships (7.4) we have the relationships - - E F - - AE - - AF - - BC , - - AB , - - AC , |E F | = |k| · |B C |, |AE | = |k| · |AB |, |AF | = |k| · |AC |. (7.7)

Taking into account k < 0 from (7.7) we derive (7.5) and (7.6). In the case k = 0 the relationship k · (a + b) = k · a + k · b reduces to the equality 0 = 0 + 0. This equality is trivially valid.


26

CHAPTER I. VECTOR ALGEBRA.

a

Exercise 7.2. Prove that k · (a + b) = k · a + k · b for the case b. For this purp ose consider the sub cases 1) a b; 3) a b and |a| = |b|; 2) a b and |a| > |b|;

4) a b and |a| < |b|.

In each of these sub cases consider two options: k > 0 and k < 0. Let's proceed to proving the distributivity of multiplication of vectors by numb ers over the addition of numb ers. The case a = 0 in the equality (k + q ) · a = k · a + q · a is trivial. In this case the equality (k + q ) · a = k · a + q · a reduces to 0 = 0 + 0. The cases k = 0 and q = 0 are also trivial. In these cases the equality (k + q ) · a = k · a + q · a reduces to the equalities q · a = 0 + q · a and k · a = k · a + 0 resp ectively. Let's consider the case a = 0 and for the sake of certainty let's assume that k > 0 and q > 0. Let's choose some arbitrary p oint A and let's build the geometric realizations of the vector k · a with the initial p oint A. Let B b e the terminal p oint of this geometric realization. Then - - AB = k · a. Similarly, we construct the geometric realization - - - - - - B C = q · a. Due to k > 0 and q > 0 the vectors AB and B C are codirected to the vector a. These vectors lie on the line AB (see Fig. 7.5). The sum of these two vectors - - - - - - AC = AB + B C (7.8)

lie on the same line and it is codirected to the vector a. The - - length of the vector AC is given by the formula |AC | = |AB | + |B C | = k |a| + q |a| = (k + q ) |a|. (7.9)


§ 7. PROPERTIES OF THE ALGEBRAIC OPERATIONS . . .

27

- - Due to AC a and due to k + q > 0 from (7.9) we derive - - AC = (k + q ) · a. (7.10)

Let's substitute (7.10) and (7.8) and take into account the re- - - - lationships AB = k · a and B C = q · a which follow from our constructions. As a result we get the required equality (k + q ) · a = k · a + q · a. Exercise 7.3. Prove that (k + q ) · a = k · a + q · a for the case where a = 0, while k and q are two numb ers of mutually opp osite signs. For the case consider the sub cases 1) |k| > |q |; 2) |k| = |q |; 3) |k| < |q |.

The associativity of the multiplication of vectors by numb ers is expressed by the equality (k q ) · a = k · (q · a). If a = 0, this equality is trivially fulfilled. It reduces to 0 = 0. If k = 0 or if q = 0, it is also trivial. In this case it reduces to 0 = 0. Let's consider the case a = 0, k > 0, and q > 0. Let's choose some arbitrary p oint A in the space E and build the geometric realization of the vector q · a with the initial p oint A. Let B b e the terminal p oint of this geometric - - realization. Then AB = q · a (see Fig. 7.6). Due to q > 0 the - - vector AB is codirected with the vector a. Let's build the vector - - - - - - AC as the product AC = k · AB = k · (q · a) relying up on the - - definition 6.1. Due to k > 0 the vector AC is also codirected - - - - with a. The lengths of AB and AC are given by the formulas |AB | = q |a|, |AC | = k |AB |. (7.11)


28

CHAPTER I. VECTOR ALGEBRA.

From the relationships (7.11) we derive the equality |AC | = k (q |a|) = (k q ) |a|. The equality - - AC = (k q ) · As a result (k q ) · a = k · (7.12)

- - (7.12) combined with AC a and k q > 0 yields - - - - a. By our construction AC = k · AB = k · (q · a). now we immediately derive the required equality (q · a).

Exercise 7.4. Prove the equality (k q ) · a = k · (q · a) in the case where a = 0, while the numb ers k and q are of opp osite signs. For this case consider the following two sub cases: 1) k > 0 and q < 0; 2) k < 0 and q > 0.

The last item 8 in the theorem 7.1 is trivial. It is immediate from the definition 6.1. § 8. Vectorial expressions and their transformations. The prop erties of the algebraic op erations with vectors listed in the theorem 7.1 are used in transforming vectorial expressions. Saying a vectorial expression one usually assumes a formula such that it yields a vector up on p erforming calculations according to this formula. In this section we consider some examples of vectorial expressions and learn some methods of transforming these expressions. Assume that a list of several vectors a1 , . . . , an is given. Then one can write their sum setting brackets in various ways: (a1 + a2 ) + (a3 + (a4 + . . . + (a
n- 1

+ an ) . . . )),
n- 1

(. . . ((((a1 + a2 ) + a3 ) + a4 ) + . . . + a

) + an ).

(8.1)

There are many ways of setting brackets. The formulas (8.1) show only two of them. However, despite the abundance of the
CopyRight c Sharipov R.A., 2010.


§ 8. VECTORIAL EXPRESSIONS AND . . .

29

ways for addition like (8.1) vectors a

brackets s (see item yield the 1 , . . . , an

etting, due to the associativity of the vector 2 in the theorem 7.1) all of the expressions same result. For this reason the sum of the can b e written without brackets at all:
n- 1

a1 + a2 + a3 + a4 + . . . + a

+ an .

(8.2)

In order to make the formula (8.2) more concise the summation sign is used. Then the formula (8.2) looks like
n

a1 + a2 + . . . + an =
i =1

ai .

(8.3)

The variable i in the formula (8.3) plays the role of the cycling variable in the summation cycle. It is called a summation index. This variable takes all integer values ranging from i = 1 to i = n. The sum (8.3) itself does not dep end on the variable i. The symb ol i in the formula (8.3) can b e replaced by any other symb ol, e. g. by the symb ol j or by the symb ol k:
n n n

ai =
i =1 j =1

aj =
k =1

ak .

(8.4)

The trick with changing (redesignating) a summation index used in (8.4) is often applied for transforming expressions with sums. The commutativity of the vector addition (see item 1 in the theorem 7.1) means that we can change the order of summands in sums of vectors. For instance, in the sum (8.2) we can write a1 + a2 + . . . + an = an + a
n- 1

+ . . . + a1 .

The most often application for the commutativity of the vector addition is changing the summation order in multiple sums. Assume that a collection of vectors aij is given which is indexed


30

CHAPTER I. VECTOR ALGEBRA.

by two indices i and j , where i = 1, . . . , m and j = 1, . . . , n. Then we have the equality
m n n m

ai j =
i =1 j =1 j =1 i =1

a

ij

(8.5)

that follows from the commutativity of the vector addition. In the same time we have the equality
m n m n

ai j =
i =1 j =1 j =1 i =1

a

ji

(8.6)

which is obtained by redesignating indices. Both methods of transforming multiple sums (8.5) and (8.6) are used in dealing with vectors. The third item in the theorem 7.1 describ es the prop erty of the null vector. This prop erty is often used in calculations. If the sum of a part of summands in (8.3) is zero, e. g. if the equality
n

a

k +1

+ . . . + an =
i =k +1

ai = 0

is fulfilled, then the sum (8.3) can b e transformed as follows:
n k

a1 + . . . + an =
i =1

ai =
i =1

ai = a1 + . . . + ak .

The fourth item in the theorem 7.1 declares the existence of an opp osite vector a for each vector a. Due to this item we can define the subtraction of vectors. Definition 8.1. The difference of two vectors a - b is the sum of the vector a with the vector b opp osite to the vector b. This fact is written as the equality a - b = a + b . (8.7)


§ 8. VECTORIAL EXPRESSIONS AND . . .

31

Exercise 8.1. Using the definitions 6.1 and 8.1, show that the opp osite vector a is produced from the vector a by multiplying it by the numb er -1, i. e. a = (-1) · a. (8.8) Due to (8.8) the vector a opp osite to a is denoted through -a and we write a = -a = (-1) · a. The distributivity prop erties of the multiplication of vectors by numb ers (see items 5 and 6 in the theorem 7.1) are used for expanding expressions and for collecting similar terms in them:
n n

·
n

a
i =1

i

=
i =1 n

· ai , i · a.

(8.9) (8.10)

i =1

i · a =

i =1

Transformations like (8.9) and (8.10) can b e found in various calculations with vectors. Exercise 8.2. Using the relationships (8.7) and (8.8), prove the following prop erties of the op eration of subtraction: a - a = 0; (a + b) - c = a + (b - c);

· (a - b) = · a - · b;

(a - b) + c = a - (b - c);

(a - b) - c = a - (b + c); ( - ) · a = · a - · a.

Here a, b, and c are vectors, while and are numb ers. The associativity of the multiplication of vectors by numb ers (see item 7 in the theorem 7.1) is used expanding vectorial expressions. Here is an example of such usage:
n n n

·

i =1

i · a

i

=
i =1

· (i · ai ) =

i =1

( i ) · ai .

(8.11)


32

CHAPTER I. VECTOR ALGEBRA.

In multiple sums this prop erty is combined with the commutativity of the regular multiplication of numb ers by numb ers:
m n n m

i =1

i ·

j =1

j · a

ij

=
j =1

j ·

i =1

i · a

ij

.

A remark. It is imp ortant to note that the associativity of the multiplication of vectors by numb ers is that very prop erty b ecause of which one can omit the dot sign in writing a product of a numb er and a vector: a = · a. Below I use b oth forms of writing for products of vectors by numb ers intending to more clarity, conciseness, and aesthetic b eauty of formulas. The last item 8 of the theorem 7.1 expresses the prop erty of the numeric unity in the form of the relationship 1 · a = a. This prop erty is used in collecting similar terms and in finding common factors. Let's consider an example: a + 3· b + 2 · a + b = a + 2 · a +3 · b + b = 1 · a + 2 · a+ + 3 · b + 1 · b = (1 + 2) · a + (3 + 1) · b = 3 · a + 4 · b. Exercise 8.3. Using the relationship 1 · a = a, prove that the conditions · a = 0 and = 0 imply a = 0. § 9. Linear combinations. Triviality, non-triviality, and vanishing. Assume that some set of n free vectors a1 , . . . , an is given. One can call it a collection of n vectors, a system of n vectors, or a family of n vectors either. Using the op eration of vector addition and the op eration of multiplication of vectors by numb ers, one can comp ose some


§ 9. LINEAR COMBINATIONS.

33

vectorial expression of the vectors a1 , . . . , an . It is quite likely that this expression will comprise sums of vectors taken with some numeric coefficients. Definition 9.1. An expression of comp osed of the vectors a1 , . . . , an tion of these vectors. The numb ers coefficients of a linear combination. I the form 1 a1 + . . . + n an is called a linear combina1 , . . . , n are called the f (9.1)

1 a1 + . . . + n an = b,

then the vector b is called the value of a linear combination. In complicated vectorial expressions linear combinations of the vectors a1 , . . . , an can b e multiplied by numb ers and can b e added to other linear combinations which can also b e multiplied by some numb ers. Then these sums can b e multiplied by numb ers and again can b e added to other sub expressions of the same sort. This process can b e rep eated several times. However, up on expanding, up on applying the formula (8.11), and up on collecting similar terms with the use of the formula (8.10) all such complicated vectorial expressions reduce to linear combinations of the vectors a1 , . . . , an . Let's formulate this fact as a theorem. Theorem 9.1. Each vectorial expression comp osed of vectors a1 , . . . , an by means of the op erations of addition and multiplication by numb ers can b e transformed to some linear combination of these vectors a1 , . . . , an . The value of a linear combination does not dep end on the order of summands in it. For this reason linear combinations differing only in order of summands are assumed to b e coinciding. For example, the expressions 1 a1 + . . . + n an and n an + . . . + 1 a1 are assumed to define the same linear combination. Definition 9.2. A linear combination 1 a1 + . . . + n an comp osed of the vectors a1 , . . . , an is called trivial if all of its coefficients are zero, i. e. if 1 = . . . = n = 0.


34

CHAPTER I. VECTOR ALGEBRA.

Definition 9.3. A linear combination 1 a1 + . . . + n an comp osed of vectors a1 , . . . , an is called vanishing or being equal to zero if its value is equal to the null vector, i. e. if the vector b in (9.1) is equal to zero. Each trivial linear combination is equal to zero. However, the converse is not valid. Noe each vanishing linear combination is trivial. In Fig. 9.1 we - - have a triangle AB C . Its sides are marked as vectors a1 = AB , - - - - a2 = B C , and a3 = C A . By construction the sum of these three vectors a1 , a2 , and a3 in Fig. 9.1 is zero: - - - - - - a1 + a2 + a3 = AB + B C + C A = 0. The equality (9.2) can b e written as follows: 1 · a1 + 1 · a2 + 1 · a3 = 0. (9.3) (9.2)

It is easy to see that the linear combination in the left hand side of the equality (9.3) is not trivial (see Definition 9.2), however, it is equal to zero according to the definition 9.3. Definition 9.4. A linear combination 1 a1 + . . . + n an comp osed of the vectors a1 , . . . , an is called non-trivial if it is not trivial, i. e. at least one of its coefficients 1 , . . . , n is not equal to zero. § 10. Linear dep endence and linear indep endence. Definition 10.1. A system of vectors a1 , . . . , an is called linearly dependent if there is a non-trivial linear combination of these vectors which is equal to zero.


§ 10. LINEAR DEPENDENCE AND . . .

35

The vectors a1 , a2 , a3 shown in Fig. 9.1 is an example of a linearly dep endent set of vectors. It is imp ortant to note that the linear dep endence is a prop erty of systems of vectors, it is not a prop erty of linear combinations. Linear combinations in the definition 10.1 are only tools for revealing the linear dep endence. It is also imp ortant to note that the linear dep endence, once it is present in a collection of vectors a1 , . . . , an , does not dep end on the order of vectors in this collection. This follows from the fact that the value of any linear combination and its triviality or non-triviality are not destroyed if we transp ose its summands. Definition 10.2. A system of vectors a1 , . . . , an is called linearly independent, if it is not linearly dep endent in the sense of the definition 10.1, i. e. if there is no linear combination of these vectors b eing non-trivial and b eing equal to zero simultaneously. One can prove the existence of a linear combination with the required prop erties in the definition 10.1 by finding an example of such a linear combination. However, proving the non-existence in the definition 10.2 is more difficult. For this reason the following theorem is used. Theorem 10.1 (linear indep endence criterion). A system of vectors a1 , . . . , an is linearly indep endent if and only if vanishing of a linear combination of these vectors implies its triviality. Proof. The proof is based on a simple logical reasoning. Indeed, the non-existence of a linear combination of the vectors a1 , . . . , an , b eing non-trivial and vanishing simultaneously means that a linear combination of these vectors is inevitably trivial whenever it is equal to zero. In other words vanishing of a linear combination of these vectors implies triviality of this linear combination. The theorem 10.1 is proved. Theorem 10.2. A system of vectors a1 , . . . , an is linearly indep endent if and only if non-triviality of a linear combination of these vectors implies that it is not equal to zero.
CopyRight c Sharipov R.A., 2010.


36

CHAPTER I. VECTOR ALGEBRA.

The theorem 10.2 is very similar to the theorem 10.1. However, it is less p opular and is less often used. Exercise 10.1. Prove the theorem 10.2 using the analogy with the theorem 10.1. § 11. Prop erties of the linear dep endence. Definition 11.1. The vector b is said to be expressed as a linear combination of the vectors a1 , . . . , an if it is the value of some linear combination comp osed of the vectors a1 , . . . , an (see (9.1)). In this situation for the sake of brevity the vector b is sometimes said to be linearly expressed through the vectors a1 , . . . , an or to be expressed in a linear way through a1 , . . . , an . There are five basic prop erties of the linear dep endence of vectors. We formulate them as a theorem. Theorem 11.1. The relation of the linear dep endence for a system of vectors p ossesses the following basic prop erties: 1) a system of vectors comprising the null vector is linearly dep endent; 2) a system of vectors comprising a linearly dep endent subsystem is linearly dep endent itself; 3) if a system of vectors is linearly dep endent, then at least one of these vectors is expressed in a linear way through other vectors of this system; 4) if a system of vectors a1 , . . . , an is linearly indep endent, while complementing it with one more vector an+1 makes the system linearly dep endent, then the vector an+1 is linearly expressed through the vectors a1 , . . . , an ; 5) if a vector b is linearly expressed through some m vectors a1 , . . . , am and if each of the vectors a1 , . . . , am is linearly expressed through some other n vectors c1 , . . . , cn , then the vector b is linearly expressed through the vectors c1 , . . . , cn .


§ 12. LINEAR DEPENDENCE FOR n = 1.

37

The prop erties 1)­5) in the theorem 11.1 are relatively simple. Their proofs are purely algebraic, they do not require drawings. I do not prove them in this b ook since the reader can find their proofs in § 3 of Chapter I in the b ook [1]. Apart from the prop erties 1)­5) listed in the theorem 11.1, there is one more prop erty which is formulated separately. Theorem 11.2 (Steinitz). If the vectors a1 , . . . , an are linearly indep endent and if each of them is linearly expressed through some other vectors b1 , . . . , bm , then m n. The Steinitz theorem 11.2 is very imp ortant in studying multidimensional spaces. We do not use it in studying the threedimensional space E in this b ook. § 12. Linear dep endence for n = 1. Let's consider the case of a system comp osed of a single a1 and apply the definition of the linear dep endence 10.1 system. The linear dep endence of such a system of one a1 means that there is a linear combination of this single which is non-trivial and equal to zero at the same time: 1 a1 = 0. vector to this vector vector

(12.1)

Non-triviality of the linear combination in the left hand side of (12.1) means that 1 = 0. Due to 1 = 0 from (12.1) we derive a1 = 0 (12.2)

(see Exercise 8.3). Thus, the linear dep endence of a system comp osed of one vector a1 yields a1 = 0. The converse prop osition is also valid. Indeed, assume that the equality (12.2) is fulfilled. Let's write it as follows: 1 · a1 = 0. (12.3)


38

CHAPTER I. VECTOR ALGEBRA.

The left hand side of the equality (12.3) combination for the system of one vector zero. Its existence means that such a sy linearly dep endent. We write this result as

is a non-trivial linear a1 which is equal to stem of one vector is a theorem.

Theorem 12.1. A system comp osed of a single vector a1 is linearly dep endent if and only if this vector is zero, i. e. a1 = 0. § 13. Linear dep endence for n = 2. Collinearity of vectors. Let's consider a system comp osed of two vectors a1 and a2 . Applying the definition of the linear dep endence 10.1 to it, we get the existence of a linear combination of these vectors which is non-trivial and equal to zero simultaneously: 1 a1 + 2 a2 = 0. (13.1)

The non-triviality of the linear combination in the left hand side of the equality (13.1) means that 1 = 0 or 2 = 0. Since the linear dep endence is not sensitive to the order of vectors in a system, without loss of generality we can assume that 1 = 0. Then the equality (13.1) can b e written as a1 = - 2 a2 . 1 (13.2)

Let's denote 2 = -2 /1 and write the equality (13.2) as a1 = 2 a2 . (13.3)

Note that the relationship (13.3) could also b e derived by means of the item 3 of the theorem 11.1. According to (13.3) , the vector a1 is produced from the vector a2 by multiplying it by the numb er 2 . In multiplying a vector by a numb er it length is usually changed (see Formulas (6.1), (6.2), (6.3), and Figs. 6.1, 6.2, and 6.3). As for its direction, it


§ 13. LINEAR DEPENDENCE FOR n = 2.

39

either is preserved or is changed for the opp osite one. In b oth of these cases the vector 2 a2 is parallel to the vector a2 . If 2 = 0m the vector 2 a2 app ears to b e the null vector. Such a vector does not have its own direction, the null vector is assumed to b e parallel to any other vector by definition. As a result of the ab ove considerations the equality (13.3) yields a
1

a2 .

(13.4)

In the case of vectors for to denote their parallelism a sp ecial term col linearity is used. Definition 13.1. Two free vectors a1 and a2 are called col linear, if their geometric realizations are parallel to some straight line common to b oth of them. As we have seen ab ove, in the case of two vectors their linear dep endence implies the collinearity of these vectors. The converse prop osition is also valid. Assume that the relationship (13.4) is fulfilled. If b oth vectors a1 and a2 are zero, then the equality (13.3) is fulfilled where we can choose 2 = 1. If at least one the two vectors is nonzero, then up to a p ossible renumerating these vectors we can assume that a2 = 0. Having - - - - built geometric realizations a2 = AB and a1 = AC , one can choose the coefficient 2 on the base of the Figs. 6.1, 6.2, or 6.3 and on the base of the formulas (6.1), (6.2), (6.3) so that the equality (13.3) is fulfilled in this case either. As for the equality (13.3) itself, we write it as follows: 1 · a1 + (-2 ) · a2 = 0. (13.5)

Since 1 = 0, the left hand side of the equality (13.5) is a nontrivial linear combination of the vectors a1 and a2 which is equal to zero. The existence of such a linear combination means that the vectors a1 and a2 are linearly dep endent. Thus, the converse prop osition that the collinearity of two vectors implies their linear dep endence is proved.


40

CHAPTER I. VECTOR ALGEBRA.

Combining the direct and converse prop ositions proved ab ove, one can formulate them as a single theorem. Theorem 13.1. A system of two vectors a1 and a2 is linearly dep endent if and only if these vectors are collinear, i. e. a1 a2 . § 14. Linear dep endence for n = 3. Coplanartity of vectors. Let;s consider a system comp osed of three vectors a1 , a2 , and a3 . Assume that it is linearly dep endent. Applying the item 3 of the theorem 11.1 to this system, we get that one of the three vectors is linearly expressed through the other two vectors. Taking into account the p ossibility of renumerating our vectors, we can assume that the vector a1 is expressed through the vectors a2 and a3 : a1 = 2 a2 + 3 a3 . (14.1)

Let A b e some arbitrary p oint of the space E. Let's build - - - - the geometric realizations a2 = AB and 2 a2 = AC . Then at the p oint C we build the geometric realizations of the vectors - - - - - - - - a3 = C D and 3 a3 = C E . The vectors AC and C E constitute two sides of the triangle AC E (see Fig. 14.1). Then the sum of - - the vectors (14.1) is presented by the third side a1 = AE . The triangle AC E is a planar form. The geometric realizations of the vectors a1 , a2 , and a3 lie on the sides of this triangle. - - Therefore they lie on the plane AC E . Instead of a1 = AE , - - - - a2 = AB , and a3 = C D by means of parallel translations we can build some other geometric realizations of these three vectors. These geometric realizations do not lie on the plane AC E , but they keep parallelism to this plane.


§ 14. LINEAR DEPENDENCE FOR n = 3.

41

Definition 14.1. Three free vectors a1 , a2 , and a3 are called coplanar if their geometric realizations are parallel to some plane common to all three of them. Lemma 14.1. The linear dep endence of three vecctors a1 , a2 , a3 implies their coplanarity. Exercise 14.1. The ab ove considerations yield a proof for the lemma 14.1 on the base of the formula (14.1) in the case where a2 = 0, a3 = 0, a2 a3 . (14.2)

Consider sp ecial cases where one or several conditions (14.2) are not fulfilled and derive the lemma 14.1 from the formula (14.1) in those sp ecial cases. Lemma 14.2. The coplanarity of three vectors a1 , a2 , a3 imply their linear dep endence. Proof. If a2 = 0 or a3 = 0, then the prop ositions of the lemma 14.2 follows from the first item of the theorem 11.1. If a2 = 0, a3 = 0, a2 a3 , then the prop osition of the lemma 14.2 follows from the theorem 13.1 and from the item 2 of the theorem 11.1. Therefore, in order to complete the proof of the lemma 14.2 we should consider the case where all of the three conditions (14.2) are fulfilled. Let A b e some arbitrary p oint of the space E. At this p oint - - we build the geometric realizations of the vectors a1 = AD , - - - - a2 = AC , and a3 = AB (see Fig. 14.2). Due to the coplanarity - - of the vectors a1 , a2 , and a3 their geometric realizations AB , - - - - AC , and AD lie on a plane. Let's denote this plane through


42

CHAPTER I. VECTOR ALGEBRA.

. Through the p oint D we draw a line parallel to the vector a3 = 0. Such a line is unique and it lies on the plane . This - - line intersects the line comprising the vector a2 = AC at some unique p oint E since a2 = 0 and a2 a3 . Considering the p oints A, E , and D in Fig. 14.2, we derive the equality - - - - - - a1 = AD = AE + E D . (14.3)

- - - - The vector AE is collinear to the vector AC = a2 = 0 since these vectors lie on the same line. For this reason there is a - - - - numb er 2 such that AE = 2 a2 . The vector E D is collinear to - - the vector AB = a3 = 0 since these vectors lie on parallel lines. - - Hence E D = 3 a3 for some numb er 3 . Up on substituting - - AE = 2 a2 , - - E D = 3 a
3

into the equality (14.3) this equality takes the form of (14.1). The last step in proving the lemma 14.2 consists in writing the equality (14.1) in the following form: 1 · a1 + (-2 ) · a2 + (-3 ) · a3 = 0. Since 1 = 0, the left han trivial linear combination to zero. The existence of the vectors a1 , a2 , and a3 d side of the of the vectors such a linear are linearly d (14.4)

equality (14.4) is a nona1 , a2 , a3 which is equal combination means that ep endent.

The following theorem is derived from the lemmas 14.1 and 14.2. Theorem 14.1. A system of three vectors a1 , a2 , a3 is linearly dep endent if and only if these vectors are coplanar. § 15. Linear dep endence for n 4.

Theorem 15.1. Any system consisting of four vectors a1 , a2 , a3 , a4 in the space E is linearly dep endent.

CopyRight c Sharipov R.A., 2010.


§ 15. LINEAR DEPENDENCE FOR n

4.

43

Theorem 15.2. Any system consisting of more than four vectors in the space E is linearly dep endent. The theorem 15.2 follows from the theorem 15.1 due to the item 3 of the theorem 11.1. Therefore it is sufficient to prove the theorem 15.1. The theorem 15.1 itself expresses a prop erty of the three-dimensional space E. Proof of the theorem 15.1. Let's choose the subsystem comp osed by three vectors a1 , a2 , a3 within the system of four vectors a1 , a2 , a3 , a4 . If these three vectors are linearly

dep endent, then in order to prove the linear dep endence of the vectors a1 , a2 , a3 , a4 it is sufficient to apply the item 3 of the theorem 11.1. Therefore in what fallows we consider the case where the vectors a1 , a2 , a3 are linearly indep endent. From the linear indep endence of the vectors a1 , a2 , a3 , according to the theorem 14.1, we derive their non-coplanarity. Moreove, from the linear indep endence of a1 , a2 , a3 due to the item 3 of the theorem 11.1 we derive the linear indep endence of any smaller subsystem within the system of these three vectors. In particular, the vectors a1 , a2 , a3 are nonzero and the vectors


44

CHAPTER I. VECTOR ALGEBRA.

a1 and a1 are not collinear (see Theorems 12.1 and 13.1), i. e. a1 = 0, a2 = 0, a3 = 0, a1 a2 . (15.1)

Let A b e some arbitrary p oint of the space E. Let's build the - - - - - - - - geometric realizations a1 = AB , a2 = AC , a3 = AD , a4 = AE with the initial p oint A. Due to the condition a1 a2 in (15.1) - - - - the vectors AB and AC define a plane (see Fig. 15.1). Let's - - denothe this plane through . The vector AD does not lie on the plane and it is not parallel to this plane since the vectors a1 , a2 , a3 are not coplanar. Let's draw a line passing through the terminal p oint of the - - vector a4 = AE and b eing parallel to the vector a3 . Since a3 , this line crosses the plane at some unique p oint F and we have - - - - - - a4 = AE = AF + F E . Now let's draw a line passing parallel to the vector a2 . Such a1 a2 this line intersects the Hence we have the following eq (15.2)

through the p oint F and b eing a line lies on the plane . Due to line AB at some unique p oint G. uality: (15.3)

- - - - - - AF = AG + GF .

- - Note that the vector AG lies on the same line as the vector - - a1 = AB . From (15.1) we get a1 = 0. Hence there is a numb er - - - - 1 such that AG = 1 a1 . Similarly, from GF a2 and a2 = 0 - - - - we derive GF = 2 a2 for some numb er 2 and from F E a3 - - and a3 = 0 we derive that F E = 3 a3 for some numb er 3 . The - - - - rest is to substitute the obtained expressions for AG , GF , and - - F E into the formulas (15.3) and (15.2) . This yields a4 = 1 a1 + 2 a2 + 3 a3 . (15.4)


§ 16. BASES ON A LINE.

45

The equality (15.4) can b e rewritten as follows: 1 · a4 + (-1 ) · a1 + (-2 ) · a2 + (-3 ) · a3 = 0. Since 1 = 0, the left hand s trivial linear combination of equal to zero. The existence that the vectors a1 , a2 , a3 theorem 15.1 is proved. (15.5)

ide of the equality (15.5) is a nonthe vectors a1 , a2 , a3 , a4 which is of such a linear combination means , a3 are linearly dep endent. The

§ 16. Bases on a line. Let a b e some line in the space E. Let's consider free vectors parallel to the line a. They have geometric realizations lying on the line a. Restricting the freedom of moving such vectors, i. e. forbidding geometric realizations outside the line a, we obtain partially free vectors lying on the line a. Definition 16.1. A system consisting of one non-zero vector e = 0 lying on a line a is called a basis on this line. The vector e is called the basis vector of this basis. Let e b e the basis vector of some basis on the line a and let x b e some other vector lying on this line (see Fig. 16.1). Then x e and hence there is a numb er x such that the vector x is expressed through a by means of the formula x = x e. (16.1)

The numb er x in the formula (16.1) is called the coordinate of the vector x in the basis e, while the formula (16.1) itself is called the expansion of the vector x in this basis. When writing the coordinates of vectors extracted from their expansions (16.1) in a basis these coordinates are usually sur-


46

CHAPTER I. VECTOR ALGEBRA.

rounded with double vertical lines x x . (16.2)

Then these coordinated turn to matrices (see [7]). The mapping (16.2) implements the basic idea of analytical geometry. This idea consists in replacing geometric ob jects by their numeric presentations. Bases in this case are tools for such a transformation. § 17. Bases on a plane. Let b e some plane in the space E. Let's consider free vectors parallel to the plane . They have geometric realizations lying on the plane . Restricting the freedom of moving such vectors, i. e. forbidding geometric realizations outside the plane , we obtain partially free vectors lying on the plane . Definition 17.1. An ordered pair of two non-collinear vectors e1 , e2 lying on a plane is called a basis on this plane. The vectors e1 and e2 are called the basis vectors of this basis. In the definition 17.1 the term "ordered system of vectors" is used. This term means a system of vectors in which some ordering of vectors is fixed: e1 is the first vector, e2 is the second vector. If we exchange the vectors e1 and e2 and take e2 for the first vector, while e1 for the second vector, that would b e an~~ other basis e1 , e2 different from the basis e1 , e2 : ~ e1 = e2 , ~ e2 = e1 .

Let e1 , e2 b e a basis on a plane and let x b e some vector


§ 17. BASES ON A PLANE.

47

lying on this place. Let's choose some arbitrary p oint O and let's build the geometric realizations of the three vectors e1 , e2 , and x with the initial p oint O: - - e1 = O A , - - e2 = O B , - - x = OC . (17.1)

Due to our choice of the vectors e1 , e2 , x and due to O the geometric realizations (17.1) lie on the plane . Let's draw a line passing through the terminal p oint of the vector x, i. e. through - - the p oint C , and b eing parallel to the vector e2 = OB . Due to non-collinearity of the vectors e1 e2 such a line intersects the - - line comprising the vector e1 = OA at some unique p oint D (see Fig. 17.1). This yields the equality - - - - - - OC = OD + DC . (17.2)

- - - - The vector OD in (17.2) is collinear with the vector e1 = OA , - - - - while the vector D C is collinear with the vector e2 = OB . For this reason there are two numb ers x1 and x2 such that - - O D = x1 e1 , - - D C = x2 e2 . (17.3)

Up on substituting (17.3) into (17.2) and taking into account the formulas (17.1) we get the equality x = x1 e1 + x2 e2 . (17.4)

The formula (17.4) is analogous to the formula (16.1). The numb ers x1 and x2 are called the coordinates of the vector x in the basis e1 , e2 , while the formula (17.4) itself is called the expansion of the vector x in this basis. When writing the coordinates of vectors they are usually arranged into columns and surrounded with double vertical lines x x x
1 2

.

(17.5)


48

CHAPTER I. VECTOR ALGEBRA.

The column of two numb ers x1 and x2 in (17.5) is called the coordinate column of the vector x. § 18. Bases in the space. Definition 18.1. An ordered system of three non-coplanar vectors e1 , e2 , e3 is called a basis in the space E. Let e1 , e2 , e3 b e a basis in the space E and let x b e some vector. Let's choose some arbitrary p oint O and build the geometric realizations of all of the four vectors e1 , e2 , e3 , and x with the common initial p oint O: - - e1 = O A , - - e3 = O C , The vectors e1 and e2 are whole system of three vectors this reason the vectors e1 = - - e2 = O B , - - x = OD . (18.1)

not collinear since otherwise the e1 , e2 , e3 would b e coplanar. For - - - - OA and e2 = OB define a plane (this plane is denoted through ~ in Fig18.1) and they lie on this - - plane. The vector e3 = OC does not lie on the plane and it is not parallel to this plane (see Fig. 18.1). Let's draw a line passing through the terminal p oint of the - - vector x = OD and b eing parallel to the vector e3 . Such a line is not parallel to the plane since e3 . It crosses the plane at some unique p oint E . As a result we get the equality - - - - - - OD = OE + E D . (18.2)


§ 18. BASES IN THE SPACE.

49

Now let's draw a line passing through the p oint E and b eing parallel to the vector e2 . The vectors e1 and e2 in (18.1) are not collinear. For this reason such a line crosses the line comprising the vector e1 at some unique p oint F . Considering the sequence of p oints O, F , E , we derive - - - - - - OE = OF + F E . Combining the equalities (18.3) and (18.2) , we obtain - - - - - - - - OD = OF + F E + E D . (18.4) (18.3)

Note that, according to our geometric construction, the following collinearity conditions are fulfilled: - - O F e1 , - - F E e2 , - - E D e3 . (18.5)

- - From the collinearity condition OF e1 in (18.5) we derive the - - existence of a numb er x1 such that OF = x1 e1 . From the other two collinearity conditions in (18.5) we derive the existence of two - - - - other numb ers x2 and x3 such that F E = x2 e2 and E D = x3 e3 . As a result the equality (18.4) is written as x = x1 e1 + x2 e2 + x3 e3 . (18.6)

The formula (18.6) is analogous to the formulas (16.1) and (17.4). The numb ers x1 , x2 , x3 are called the coordinates of the vector x in the basis e1 , e2 , e3 , while the formula (18.6) itself is called the expansion of the vector x in this basis. In writing the coordinates of a vector they are usually arranged into a column surrounded with two double vertical lines: x x x x
1 2 3

.

(18.7)

CopyRight c Sharipov R.A., 2010.


50

CHAPTER I. VECTOR ALGEBRA.

The column of the numb ers x1 , x2 , x3 in (18.7) is called the coordinate column of a vector x. § 19. Uniqueness of the expansion of a vector in a basis. Let e1 , e2 , e3 b e some basis in the space E. The geometric construction shown in Fig. 18.1 can b e applied to an arbitrary vector x. It yields an expansion of (18.6) of this vector in the basis e1 , e2 , e3 . However, this construction is not a unique way foe expanding a vector in a basis. For example, instead of the plane OAB the plane OAC can b e taken for , while the line can b e directed parallel to the vector e2 . The the line E F is directed parallel to the vector e3 and the p oint F is obtained in its intersection with the line OA comprising the geometric realization of the vector e1 . Such a construction p otentially could yield some other expansion of a vector x in the basis e1 , e2 , e3 , i. e. an expansion with the coefficients different from those of (18.6). The fact that actually this does not happ en should certainly b e proved. Theorem 19.1. The expansion of an arbitrary vector x in a given basis e1 , e2 , e3 is unique. Proof. The proof is by contradiction. Assume that the expansion (18.6) is not unique and we have some other expansion of the vector x in the basis e1 , e2 , e3 : x = x1 e1 + x2 e2 + x3 e3 . ~ ~ ~ (19.1)

Let's subtract th expansion (18.6) from the expansion (19.1). Up on collecting similar term we get (x1 - x1 ) e1 + (x2 - x2 ) e2 + (x3 - x3 ) e3 = 0. ~ ~ ~ (19.2)

According to the definition 18.1, the basis e1 , e2 , e3 is a triple of non-coplanar vectors. Due to the theorem 14.1 such a triple of vectors is linearly indep endent. Looking at (19.2), we see that


§ 20. INDEX SETTING CONVENTION.

51

there we have a linear combination of the basis vectors e1 , e2 , e3 which is equal to zero. Applying the theorem 10.1, we conclude that this linear combination is trivial, i. e. x1 - x1 = 0, ~ x2 - x2 = 0, ~ x3 - x3 = 0. ~ (19.3)

The equalities (19.3) mean that the coefficients in the expansions (19.1) and (18.6) do coincide, which contradicts the assumption that these expansions are different. The contradiction obtained proves the theorem 19.1. Exercise 19.1. By analogy to the theorem 19.1 formulate and prove uniqueness theorems for expansions of vectors in bases on a plane and in bases on a line. § 20. Index setting convention. The theorem 19.1 on the uniqueness of an expansion of vector in a basis means that the mapping (18.7), which associates vectors with their coordinates in some fixed basis, is a bijective mapping. This makes bases an imp ortant tool for quantitative description of geometric ob jects. This tool was improved substantially when a sp ecial index setting convention was admitted. This convention is known as Einstein's tensorial notation. Definition 20.1. The index setting convention, which is also known as Einstein's tensorial notation, is a set of rules for placing indices in writing comp onents of numeric arrays representing various geometric ob jects up on choosing some basis or some coordinate system. Einstein's tensorial notation is not a closed set of rules. When new typ es of geometric ob jects are designed in science, new rules are added. For this reason b elow I formulate the index setting rules as they are needed.


52

CHAPTER I. VECTOR ALGEBRA.

Definition 20.2. Basis vectors in a basis are enumerated by lower indices, while the coordinates of vectors expanded in a basis are enumerated by upp er indices. The rule formulated in the definition 20.2 b elongs to Einstein's tensorial notation. According to this rule the formula (18.6) should b e rewritten as
3

x = x e1 + x e2 + x e3 =
i =1

1

2

3

xi ei ,

(20.1)

whire the mapping (18.7) should b e written in the following way: x x x x
1 2 3

.

(20.2)

Exercise 20.1. The rule from the definition 20.2 is applied for bases on a line and for bases on a plane. Relying on this rule rewrite the formulas (16.1), (16.2), (17.4), and (17.5). § 21. Usage of the coordinates of vectors. Vectors can b e used in solving various where the basic algebraic op erations with These are the op eration of vector addition multiplication of vectors by numb ers. Th coordinates of vectors in bases relies on the geometric problems, them are p erformed. and the op eration of e usage of bases and following theorem.

Theorem 21.1. Let some basis e1 , e2 , e3 in the space E b e chosen and fixed. In this situation when adding vectors their coordinates are added, while when multiplying a vector by a numb er its coordinates are multiplied by this numb er, i. e. if x x x
1 2 3

x

,

y

y y y

1 2 3

,

(21.1)


§ 2 2 . C H A N G I N G A B A SI S.

53

then for x + y and · x we have the relationships x+y x1 + y x2 + y x3 + y
1 2 3

,

·x

x x x

1 2 3

.

(21.2)

Exercise 21.1. Prove the theorem 21.1, using the formulas (21.1) and (21.2) for this purp ose, and prove the theorem 19.1. Exercise 21.2. Formulate and prove theorems analogous to the theorem 21.1 in the case of bases on a line and on a plane. § 22. Changing a basis. Transition formulas and transition matrices. When solving geometric problems sometimes one needs to change a basis replacing it with another basis. Let e1 , e2 , e3 and ~~~ e1 , e2 , e3 b e two bases in the space E. If the basis e1 , e2 , e3 ~~~ is replaced by the basis e1 , e2 , e3 , then e1 , e2 , e3 is usually ~~~ called the old basis, while e1 , e2 , e3 is called the new basis. The procedure of changing an old basis for a new one can b e understood as a transition from an old basis to a new basis or, in other words, as a direct transition. Conversely, changing a new basis for an old one is understood as an inverse transition. ~~~ Let e1 , e2 , e3 and e1 , e2 , e3 b e two bases in the space E, ~~~ where e1 , e2 , e3 is an old basis and e1 , e2 , e3 is a new basis. In the direct transition procedure vectors of a new basis are expanded in an old basis, i. e. we have
3 2 1 ~ e1 = S 1 e1 + S 1 e2 + S 1 e3 , 1 2 3 ~ e2 = S 2 e1 + S 2 e2 + S 2 e3 , 1 2 3 ~ e3 = S 3 e1 + S 3 e2 + S 3 e3 .

(22.1)

The formulas (22.1) are called the direct transition formulas. The 1 2 3 numeric coefficients S1 , S1 , S1 in (22.1) are the coordinates


54

CHAPTER I. VECTOR ALGEBRA.

~ of the vector e1 expanded in the old basis. According to the definition 20.2, they are enumerated by an upp er index. The ~ lower index 1 of them is the numb er of the vector e1 of which they are the coordinates. It is used in order to distinguish the ~ ~ ~ coordinates of the vector e1 from the coordinates of e2 and e3 in the second and in the third formulas (22.1). Let's apply the first mapping (20.2) to the transition formulas ~~~ and write the coordinates of the vectors e1 , e2 , e3 as columns: S ~ e1 S S
1 1 2 1 3 1

S , ~ e2 S S

1 2 2 2 3 2

S , ~ e3 S S

1 3 2 3 3 3

.

(22.2)

The columns (22.2) are usually glued into a single matrix. Such a matrix is naturally denoted through S : S S= S S
1 1 2 1 3 1

S S S

1 2 2 2 3 2

S S S

1 3 2 3 3 3

.

(22.3)

Definition 22.1. The matrix (22.3) whose comp onents are determined by the direct transition formulas (22.1) is called the direct transition matrix.
i Note that the comp onents of the direct transition matrix Sj are enumerated by two indices one of which is an upp er index, while the other is a lower index. These indices define the p osition i of the element Sj in the matrix S : the upp er index i is a row numb er, while the lower index j is a column numb er. This notation is a part of a general rule.

Definition 22.2. If enumerated by indices a matrix of these elem numb er, while the lower

elements of a double index array are on different levels, then in comp osing ents the upp er index is used as a row index is used a column numb er.


§ 2 2 . C H A N G I N G A B A SI S.

55

Definition enumerated by matrix of these while the secon

22.3. If elements of a double index array are indices on the same level, then in comp osing a elements the first index is used as a row numb er, d index is used a column numb er. of gh of a

The definitions 22.2 and 22.3 can b e considered as a part the index setting convention from the definition 20.1, thou formally they are not since they do not define the p ositions array indices, but the way to visualize the array as a matrix. The direct transition formulas (22.1) can b e written in concise form using the summation sign for this purp ose:
3

~ ej =
i =1

i Sj ei , where j = 1, 2, 3.

(22.4)

There is another way to write the formulas (22.1) concisely. It is based on the matrix multiplication (see [7]): S ~ e
1 1 1 2 1 3 1

S S S

~ e

2

~ e

3

=e

1

e

2

e

3

·S S

1 2 2 2 3 2

S S S

1 3 2 3 3 3

.

(22.5)

~~~ Note that the basis vectors e1 , e2 , e3 and e1 , e2 , e3 in the formula (22.5) are written in rows. This fact is an instance of a general rule. Definition 22.4. If elements of a single index array are enumerated by lower indices, then in matrix presentation they are written in a row, i. e. they constitute a matrix whose height is equal to unity. Definition 22.5. If elements of a single index array are enumerated by upp er indices, then in matrix presentation they are written in a column, i. e. they constitute a matrix whose width is equal to unity.


56

CHAPTER I. VECTOR ALGEBRA.

Note that writing the comp onents of a vector x as a column in the formula (20.2) is concordant with the rule from the definition 22.5, while the formula (18.7) violates two rules at once -- the rule from the definition 20.2 and the rule from the definition 22.5. Now let's consider the inverse transition from the new basis ~1 , e2 , e3 to the old basis e1 , e2 , e3 . In the inverse transition e~~ procedure the vectors of an old basis are expanded in a new basis:
1 2 3 ~ ~ ~ e1 = T1 e1 + T1 e2 + T1 e3 , 3 2 1 ~ ~ ~ e2 = T2 e1 + T2 e2 + T2 e3 ,

(22.6)

e3 =

1 T3

~ e1 +

2 T3

~ e2 +

3 T3

~ e3 .

The formulas (22.6) are called the inverse transition formulas. The numeric coefficients in the formulas (22.6) are coordinates of the vectors e1 , e2 , e3 in their expansions in the new basis. These coefficients are arranged into columns:
1 T1 1 T2 2 e2 T2 , 3 T2 1 T3

e1

2 T1 , 3 T1

e3

2 T3 . 3 T3

(22.7)

Then the columns (22.7) are united into a matrix:
1 T1 1 T2 2 T2 3 T2 1 T3 2 T3 . 3 T3

T=

2 T1 3 T1

(22.8)

Definition 22.6. The matrix (22.8) whose comp onents are determined by the inverse transition formulas (22.6) is called the inverse transition matrix. The inverse transition formulas (22.6) have a concise form, analogous to the formula (22.4):
3

ej =
i =1

~ Tji ei , where j = 1, 2, 3.

(22.9)

CopyRight c Sharipov R.A., 2010.


§ 23. SOME INFORMATION . . .

57

There is also a matrix form of the formulas (22.6):
1 T1 1 T2 2 T2 3 T2 1 T3 2 T3 . 3 T3

e

1

e

2

e

3

~ =e

1

~ e

2

~ e

3

2 · T1 3 T1

(22.10)

The formula (22.10) is analogous to the formula (22.5). Exercise 20.1. By analogy to (22.1) and (22.6) write the transition formulas for bases on a plane and for bases on a line (see Definitions 16.1 and 17.1). Write also the concise and matrix versions of these formulas. § 23. Some information on transition matrices. Theorem 23.1. The matrices S and T whose comp onents are determined by the transition formulas (22.1) and (22.6) are inverse to each other, i. e. T = S -1 and S = T -1 . I do not prove the theorem 23.1 in this b ook. The reader can find this theorem and its proof in [1]. The relationships T = S -1 and S = T -1 from the theorem 23.1 mean that the product of S by T and the product of T by S b oth are equal to the unit matrix (see [7]): S·T =1 T · S = 1. (23.1)

Let's recall that the unit matrix is a square n â n matrix that has ones on the main diagonal and zeros in all other p ositions. Such a matrix is often denoted through the same symb ol 1 as the numeric unity. Therefore we can write 100 010. 001

1=

(23.2)


58

CHAPTER I. VECTOR ALGEBRA.

In order to denote the comp onents of the unit matrix (23.2) the symb ol is used. The indices enumerating rows and columns can b e placed either on the same level or on different levels:
ij i = j = ij =

1 for i = j, 0 for i = j.

(23.3)

Definition 23.1. The double index numeric array determined by the formula (23.3) is called the Kronecker symbol 1 or the Kronecker delta. The p ositions of indices in the Kronecker symb ol are determined by a context where it is used. For example, the relationships (23.1) can b e written in comp onents. In this particular case the indices of the Kronecker symb ol are placed on different levels:
3 ij i Sj Tk = k , j =1 j =1 3 j i Tji Sk = k .

(23.4)

Such a placement of the indices in the Kronecker symb ol in (23.4) is inherited from the transition matrices S and T . Noter that the transition matrices S and T are square matrices. For such matrices the concept of the determinant is introduced (see [7]). This is a numb er calculated through the comp onents of a matrix according to some sp ecial formulas. In the case of the unit matrix (23.2) these formulas yield det 1 = 1. (23.5)

The following fact is also well known. Its proof can b e found in the b ook [7]. Theorem 23.2. The determinant of a product of matrices is equal to the product of their determinants.
1

Don't mix with the Kronecker symbol used in number theory (see [8]).


§ 2 4 . I N DE X S E T T I N G I N S U M S .

59

Let's apply the theorem 23.2 and the formula (23.5) to the products of the matrices S and T in (23.1). This yields det S · det T = 1. (23.6)

Definition 23.2. A matrix with zero determinant is called degenerate. If the determinant of a matrix is nonzero, such a matrix is called non-degenerate. From the formula (23.6) and the definition 23.2 we immediately derive the following theorem. Theorem 23.3. For any two bases in the space E the corresp onding transition matrices S and T are non-degenerate and the product of their determinants is equal to the unity. Theorem 23.4. Each non-degenerate 3 â 3 matrix S is a transition matrix relating some basis e1 , e2 , e3 in the space E with ~~~ some other basis e1 , e2 , e3 in this space. The theorem 23.4 is a strengthened version of the theorem 23.3. Its proof can b e found in the b ook [1]. Exercise 23.1. Formulate theorems analogous to the theorems 23.1, 23.2, and 23.4 in the case of bases on a plane and in the case of bases on a line. § 24. Index setting in sums. As we have already seen, formulas with sums arise. in a concise form using th this way one should follow definitions. in dealing with coordinates of vectors It is convenient to write these sums e summation sign. Writing sums in some rules, which are listed b elow as

Definition 24.1. Each summation sign in a formula has its scop e. This scop e b egins immediately after the summation sign to the right of it and ranges up to some delimiter: 1) the end of the formula;


60

CHAPTER I. VECTOR ALGEBRA.

2) the equality sign; 3) the plus sign «+» or the minus sign «-» not enclosed into brackets op ened after a summation sign in question; 4) the closing bracket whose op ening bracket precedes the summation sign in question. Let's recall that a summation sign is present in a formula and if some variable is used as a cycling variable in this summation sign, such a variable is called a summation index (see Formula (8.3) and the comment to it). Definition 24.2. Each summation index can b e used only within the scop e of the corresp onding summation sign. Apart from simple sums, multiple sums can b e used in formulas. They ob ey the following rule. Definition 24.3. A variable cannot b e used as a summation index in more than one summation signs of a multiple sum. Definition 24.4. Variables which are not summation indices are called free variables. Summation indices as well as free variables can b e used as indices enumerating basis vectors and array comp onents. The following terminology goes along with this usage. Definition 24.5. A free variable which is used as an index is called a free index. In the definitions 24.1, 24.2 and 24.3 the commonly admitted rules are listed. Apart from them there are more sp ecial rules which are used within the framework of Einstein's tensorial notation (see Definition 20.1). Definition 24.6. If an expression is a simple sum or ple sum and if each summand of it does not comprise oth then each free index should have exactly one entry in pression, while each summation index should enter twice as an upp er index and once as a lower index. a multier sums, this ex-- once


§ 2 4 . I N DE X S E T T I N G I N S U M S .

61

Definition 24.7. The expression built according to the definition 24.6 can b e used for comp osing sums with numeric coefficients. Then all summands in such sums should have the same set of free indices and each free index should b e on the same level (upp er or lower) in all summands. Regardless to the numb er of summands, in counting the numb er of entries to the whole sum each free index is assumed to b e entering only once. The level of a free index in the sum (upp er or lower) is determined by its level in each summand. Lets consider some expressions as examples:
3 3 3 3

ai bi ,
i =1 3 i =1 3

ai gij ,
i =1 k =1 3 3

ai bk gik , -
r Cq s B r =1 s=1 ism

(24.1) vr . (24.2)

2
k =1

Ai bm k

u vq +
j =1

k

3C

ij m jq

Exercise 24.1. Verify that each expression in (24.1) satisfies the definition 24.6, while the expression (24.2) satisfies the definition 24.7. Definition 24.8. Sums comp osed according to the definition 24.7 can enter as sub expressions into simple and multiple sums which will b e external sums with resp ect to them. Then some of their free indices or all of their free indices can turn into summation indices. Those of free indices that remain free are included into the list of free indices of the whole expression. In counting the numb er of entries of an index in a sum included into an external simple or multiple sum the rule from the definition 24.7 is applied. Taking into account this rue, each free index of the ultimate expression should enter it exactly once, while each summation index should enter it exactly twice -- once as an upp er index and once as a lower index. In counting the numb er of entries of an index in a sum included into an


62

CHAPTER I. VECTOR ALGEBRA.

external simple or multiple sum the rule from the definition 24.7 is applied. Taking into account this rue, each free index of the ultimate expression should enter it exactly once, while each summation index should enter it exactly twice -- once as an upp er index and once as a lower index. In counting the numb er of entries of an index in a sum included into an external simple or multiple sum the rule from the definition 24.7 is applied. Taking into account this rue, each free index of the ultimate expression should enter it exactly once, while each summation index should enter it exactly twice -- once as an upp er index and once as a lower index. In counting the numb er of entries of an index in a sum included into an external simple or multiple sum the rule from the definition 24.7 is applied. Taking into account this rue, each free index of the ultimate expression should enter it exactly once, while each summation index should enter it exactly twice -- once as an upp er index and once as a lower index. As an example we consider the following expression which comprises inner and outer sums:
3 3

Ai k
k =1

B+
i =1

k q

k Ciq ui .

(24.3)

Exercise 24.2. Make sure that the expression (24.3) satisfies the definition 24.8. Exercise 24.3. Op en the brackets in (24.3) and verify that the resulting expression satisfies the definition 24.7. The expressions built according to the definitions 24.6, 24.7, and 24.8 can b e used for comp osing equalities. In comp osing equalities the following rule is applied. Definition 24.9. Both sides of an equality should have the same set of free indices and each free index should have the same p osition (upp er or lower) in b oth sides of an equality.


§ 25. TRANSFORMATION OF THE COORDINATES . . .

63

§ 25. Transformation of the coordinates of vectors under a change of a basis. ~~~ Let e1 , e2 , e3 and e1 , e2 , e3 b e two bases in the space E and ~~~ let e1 , e2 , e3 b e changed for the basis e1 , e2 , e3 . As we already mentioned, in this case the basis e1 , e2 , e3 is called an old basis, ~~~ while e1 , e2 , e3 is called a new basis. Let's consider some arbitrary vector x in the space E. Expanding this vector in the old basis e1 , e2 , e3 and in the new ~~~ basis e1 , e2 , e3 , we get two sets of its coordinates: x x x x
1 2 3

,

x ~ x x ~ x ~

1 2 3

.

(25.1)

Both mappings (25.1) are bijective. For this reason there is a bijective corresp ondence b etween two sets of numb ers x1 , x2 , x3 and x1 , x2 , x3 . In order to get explicit formulas expressing ~~~ the coordinates of the vector x in the new basis through its coordinates in the old basis we use the expansion (20.1) :
3

x=
j =1

x j ej .

(25.2)

Let's apply the inverse transition formula (22.9) in order to express the vector ej in (25.2) through the vectors of the new ~~~ basis e1 , e2 , e3 . Up on substituting (22.9) into (25.2), we get
3 3

x=
j =1 3 3

x

j i =1 3

~ Tji e

i

= (25.3) Tji x
j

3

=
j =1 i =1

~ x j Tji ei =
i =1 j =1

~ ei .

The formula (25.3) expresses the vector x as a linear combination ~~~ of the basis vectors e1 , e2 , e3 , i. e. it is an expansion of the vector
CopyRight c Sharipov R.A., 2010.


64

CHAPTER I. VECTOR ALGEBRA.

x in the new basis. Due to the uniqueness of the expansion of a vector in a basis (see Theorem 19.1) the coefficients of such an expansion should coincide with the coordinates of the vector x in ~~~ the new basis e1 , e2 , e3 :
3

xi = ~
j =1

Tji x j , there i = 1, 2, 3.

(25.4)

The formulas (25.4) expressing the coordinates of an arbitrary vector x in a new basis through its coordinates in an old basis are called the direct transformation formulas. Accordingly, the formulas expressing the coordinates of an arbitrary vector x in an old basis through its coordinates in a new basis are called the inverse transformation formulas. The latter formulas need not b e derived separately. It is sufficient to move the tilde sign from the left hand side of the formulas (25.4) to their right hand side i and replace Tji with Sj . This yields
3

x=
j =1

i

i Sj x j , where i = 1, 2, 3. ~

(25.5)

The direct transformation formulas (25.4) have the expanded form where the summation is p erformed explicitly:
1 1 1 x1 = T1 x1 + T2 x2 + T3 x3 , ~ 2 2 2 x2 = T1 x1 + T2 x2 + T3 x3 , ~ 3 3 3 x3 = T1 x1 + T2 x2 + T3 x3 . ~

(25.6)

The same is true for the inverse transformation formulas (25.5):
1 1 1 x1 = S 1 x1 + S 2 x2 + S 3 x3 , ~ ~ ~ 2 2 2 x2 = S 1 x1 + S 2 x2 + S 3 x3 , ~ ~ ~

(25.7)

x = S x +S x +S x . ~ ~ ~

3

3 1

1

3 2

2

3 3

3


§ 26. SCALAR PRODUCT.

65

Along with (25.6), there is the matrix form of the formulas (25.4): x ~ x ~ x ~
1 2 3 1 T1 1 T2 2 T2 3 T2 1 T3 2 T3 3 T3

=

2 T1 3 T1

x ·x x

1 2 3

(25.8)

Similarly, the inverse transformation formulas (25.5), along with the expanded form (25.7), have the matrix form either x x x Exercise mulas (25.4), plane and for on a plane an
1 2 3 1 T1 1 T2 2 T2 3 T2 1 T3 2 T3 3 T3

=

2 T1 3 T1

x ~ ~ ·x x ~

1 2 3

(25.9)

25.1. Write the analogs of the transformation for(25.5) , (25.6), (25.7), (25.8), (25.9) for vectors on a vectors on a line expanded in corresp onding bases d on a line. § 26. Scalar product.

Let a and b b e two nonzero free vectors. Let's build their - - - - geometric realizations a = OA and b = OB at some arbitrary p oint O. The smaller of two angles formed by the rays [OA) and [OB ) at the p oint O is called the angle between - - - - vectors OA and OB . In Fig. 26.1 this angle is denoted through . The value of the angle ranges from 0 to : 0 .

- - The lengths of the vectors OA and - - OB do not dep end on the choice of a p oint O (see Definitions 3.1 and 4.2). The same is true for the angle b etween them. Therefore


66

CHAPTER I. VECTOR ALGEBRA.

we can deal with the lengths of the free vectors a and b and with the angle b etween them: |a| = |OA|, |b| = |OB |, ab = AOB = .

In the case where a = 0 or b = 0 the lengths of the vectors a and b are defined, but the angle b etween these vectors is not defined. Definition 26.1. The scalar product of two nonzero vectors a and b is a numb er equal to the product of their lengths and the cosine of the angle b etween them: (a, b) = |a| |b| cos . (26.1)

In the case where a = 0 or b = 0 the scalar product (a, b) is assumed to b e equal to zero by definition. A comma is the multiplication sign in the writing the scalar product, not by itself, but together with round brackets surrounding the whole expression. These brackets are natural delimiters for multiplicands: the first multiplicand is an expression b etween the op ening bracket and the comma, while the second multiplicand is an expression b etween the comma and the closing bracket. Therefore in complicated expressions no auxiliary delimiters are required. For example, in the formula (a + b, c + d) the sums a + b and c + d are calculated first, then the scalar multiplication is p erformed. A remark. Often the scalar product is written as a · b. Even the sp ecial term «dot product» is used. However, to my mind, this notation is not good. It is misleading since the dot sign is used for denoting the product of a vector and a numb er and for denoting the product of two numb ers.


§ 27. ORTHOGONAL PROJECTION ONTO A LINE.

67

§ 27. Orthogonal pro jection onto a line. Let a and b b e two free vectors such that a = 0. Let's - - - - build their geometric realizations a = OA and b = OB at some arbitrary p oint O. The nonzero - - vector OA defines a line. Let's drop the p erp endicular from the - - terminal p oint of the vector OB , i. e. from the p oint B , to this line and let's denote through C the base of this p erp endicular (see Fig. 27.1). In the sp ecial case where b a and where the p oint B lies on the line OA we choose the p oint C coinciding with the p oint B . - - - - The p oint C determines two vectors OC and C B . The - - - - vector OC is collinear to the vector a, while the vector C B is p erp endicular to it. By means of parallel translations one can - - - - replicate the vectors OC and C B up to free vectors b and b resp ectively. Note that the p oint C is uniquely determined by the p oint B and by the line OA (see Theorem 6.5 in Chapter I I I of the b ook [7]). For this reason the vectors b and b do not dep end on the choice of a p oint O and we can formulate the following theorem. Theorem 27.1. For any nonzero vector a = 0 and for any vector b there two unique vectors b and b such that the vector b is collinear to a, the vector b is p erp endicular to a, and they b oth satisfy the equality b eing the expansion of the vector b: b = b + b . (27.1)

One should recall the sp ecial case where the p oint C coincides with the p oint B . In this case b = 0 and we cannot verify visually the orthogonality of the vectors b and a. In order


68

CHAPTER I. VECTOR ALGEBRA.

to extend the theorem 27.1 to this sp ecial case the following definition is introduced. Definition 27.1. All null vectors are assumed to b e p erp endicular to each other and each null vector is assumed to b e p erp endicular to any nonzero vector. Like the definition 3.2, the definition 27.1 is formulated for geometric vectors. Up on passing to free vectors it is convenient to unite the definition 3.2 with the definition 27.1 and then formulate the following definition. Definition 27.2. A free null vector 0 of any physical nature is codirected to itself and to any other vector. A free null vector 0 of any physical nature is p erp endicular to itself and to any other vector. When taking into account the definition 27.2, the theorem 27.1 is proved by the constructions preceding it, while the expansion (27.1) follows from the evident equality - - - - - - OB = OC + C B . Assume that a vector a = 0 is fixed. In this situation the theorem 27.1 provides a mapping a that associates each vector b with its parallel comp onent b . Definition 27.3. The mapping a vector b with its parallel comp onent b is called the orthogonal projection onto a = 0 or, more exactly, the orthogonal tion of the vector a = 0. that associates in the expans a line given by projection onto each free ion (27.1) the vector the direc-

The orthogonal pro jection a is closely related to the scalar product of vectors. This relation is established by the following theorem. Theorem 27.2. For each nonzero vector a = 0 and for any


§ 27. ORTHOGONAL PROJECTION ONTO A LINE.

69

vector b the vector a (b) is calculated by means of the formula a (b) = (b, a) a. |a|2 (27.2)

Proof. If b = 0 b oth sides of the equality (27.2) do vanish and it is trivially fulfilled. Therefore we can assume that b = 0. It is easy to see that the vectors in two sides of the equality (27.2) are collinear. For the b eginning let's prove that the lengths of these two vectors are equal to each other. The length of the vector a (b) is calculated according to Fig. 27.1: |a (b)| = |b | = |b| | cos |. (27.3)

The length of the vector in the right hand side of the formula (27.2) is determined by the formula itself: (b, a) |b| |a| | cos | |(b, a)| = |b| | cos |. a= |a| = 2 2 |a| |a| |a| (27.4)

Comparing the results of (27.3) and (27.4), we conclude that |a (b)| = (b, a) a. |a|2 (27.5)

Due to (27.5) in order to prove the equality (27.2) it is sufficient to prove the codirectedness of vectors a (b) (b, a) a. |a|2 (27.6)

Since a (b) = b , again applying Fig. 27.1, we consider the following three p ossible cases: 0 < /2, = /2, /2 < .


70

CHAPTER I. VECTOR ALGEBRA.

In the first case b oth vectors (27.6) are codirected with the vector a = 0. Hence they are codirected with each other. In the second case b oth vectors (27.6) are equal to zero. They are codirected according to the definition 3.2. In the third case b oth vectors (27.6) are opp osite to the vector a = 0. Therefore they are again codirected with each other. The relationship (27.6) and the theorem 27.2 in whole are proved. Definition 27.4. A mapping f acting from the set of all free vectors to the set of all free vectors is called a linear mapping if it p ossesses the following two prop erties: 1) f (a + b) = f (a) + f (b); 2) f ( a) = f (a). The prop erties 1) and 2), which should b e fulfilled for any two vectors a and b and for any numb er , constitute a prop erty which is called the linearity. Theorem 27.3. For any nonzero vector a = 0 the orthogonal pro jection a onto a line given by the vector a is a linear mapping. In order to prove the theorem 27.3 we need the following auxiliary lemma. Lemma 27.1. For any nonzero vector a = 0 the sum of two vectors collinear to a is a vector collinear to a and the sum of two vectors p erp endicular to a is a vector p erp endicular to a. Proof of the lemma 27.1. The first prop osition of the lemma is obvious. It follows immediately from the definition 5.1. Let's prove the second prop osition. Let b and c b e two vectors, such that b a and c a.
CopyRight c Sharipov R.A., 2010.


§ 27. ORTHOGONAL PROJECTION ONTO A LINE.

71

In the cases b = 0, c = 0, b + c = 0, and in the case b c the second prop osition of the lemma is also obvious. In these cases geometric realizations of all the three vectors b, c, and b + c can b e chosen such that they lie on the same line. Such a line is p erp endicular to the vector a. Let's consider the case where b c. Let's build a geometric - - realization of the vector b = OB with initial p oint at some - - arbitrary p oint O. Then we lay the vector c = B C at the - - terminal p oint B of the vector OB . Since b c, the p oints O, B , and C do not lie on a single straight line altogether. Hence they determine a plane. We denote this plane through . The sum - - - - - - of vectors OC = OB + B C lies on this plane. It is a geometric - - realization for the vector b + c, i.. e. OC = b + c. - - - - We build two geometric realizations OC and B D for the vector a. These are two different geometric vectors. From the - - - - equality OA = B D we derive that the lines OA and B D are parallel. Due to b a and c a the line B D is p erp endicular to the pair of crosswise intersecting lines OB and B C lying on the plane . Hence it is p erp endicular to this plane. From B D and B D OA we derive OA , while from OA we derive - - - - OA OC . Hence the sum of vectors b + c is p erp endicular to the vector a. The lemma 27.1 is proved. Proof of the theorem 27.3. According to the definition 27.4, in order to prove the theorem we need to verify two linearity conditions for the mapping a . The first of these conditions in our particular case is written as the equality a (b + c) = a (b) + a (c). (27.7)

Let's denote d = b + c. According to the theorem 27.2, there are expansions of the form (27.1) for the vectors b, c, and d: b = b + b , c = c + c , (27.8) (27.9)


72

CHAPTER I. VECTOR ALGEBRA.

d = d + d .

(27.10)

According the same theorem 27.2, the comp onents of the expansions (27.8), (27.9), (27.10) are uniquely fixed by the conditions b c d a, a, a, b a , (27.11) (27.12) (27.13)

d a .

c a,

Adding the equalities (27.8) and (27.9) , we get d = b + c = (b + c ) + (b + c ). (27.14)

Due to (27.11) and (27.12) we can apply the lemma 27.1 to the comp onents of the expansion (27.14) . This yields (b + c ) a, (b + c ) a. (27.15)

The rest is to compare (27.14) with (27.10) and (27.15) with (27.13). From this comparison, applying the theorem 27.2, we derive the following relationships: d =b +c , d = b + c . (27.16)

According to the definition 27.3, the first of the ab ove relationships (27.16) is equivalent to the equality (27.7) which was to b e verified. Let's proceed to proving the second linearity condition for the mapping a . It is written as follows: a ( b) = a (b). (27.17)

Let's denote e = b and then, applying the theorem 27.2, write b = b + b , e = e + e . (27.18) (27.19)


§ 28. PROPERTIES OF THE SCALAR PRODUCT.

73

According to the theorem 27.2, the comp onents of the expansions (27.18) and (27.19) are uniquely fixed by the conditions b e a, a, b a , e a . (27.20) (27.21)

Let's multiply b oth sides of (27.18) by . Then we get e = b = b + b . (27.22)

Multiplying a vector by the numb er , we get a vector collinear to the initial vector. For this reason from (27.20) we derive ( b ) a, ( b ) a, (27.23)

Let's compare (27.22) with (27.19) and (27.23) with (27.21). Then, applying the theorem 27.2, we obtain e = b , e = b . (27.24)

According to the definition 27.3 the first of the equalities (27.24) is equivalent to the required equality (27.17) . The theorem 27.3 is proved. § 28. Prop erties of the scalar product. Theorem 28.1. The following four prop erties a, b, c and for any numb 1) (a, b) = (b, a); 2) (a + b, c) = (a, c) + 3) ( a, c) = (a, c); 4) (a, a) 0 and (a, a) scalar product of vectors p ossesses the which are fulfilled for any three vectors er : (b, c); = 0 implies a = 0.


74

CHAPTER I. VECTOR ALGEBRA.

De called called cand;

finition 28.1. The prop erty 1) in the theorem 28.1 is the prop erty of symmetry; the prop erties 2) and 3) are the prop erties of linearity with respect to the first multiplithe prop erty 4) is called the prop erty of positivity .

Proof of the theorem 28.1. The prop erty of symmetry 1) is immediate from the definition 26.1 and the formula (26.1) in this definition. Indeed, if one of the vectors a or b is equal to zero, then b oth sides of the equality (a, b) = (b, a) equal to zero. Hence the equality is fulfilled in this case. In the case of the nonzero vectors a and b the angle is determined by the pair of vectors a and b according to Fig. 26.1, it does not dep end on the order of vectors in this pair. Therefore the equality (a, b) = (b, a) in this case is reduced to the equality |a| |b| cos = |b| |a| cos . It is obviously fulfilled since |a| and |b| are numb ers complemented by some measure units dep ending on the physical nature of the vectors a and b. Let's consider the prop erties of linearity 2) and 3). If c = 0, then b oth sides of the equalities (a + b, c) = (a, c) + (b, c) and ( a, b) = (a, b) are equal to zero. Hence these equalities are fulfilled in this case. If c = 0, then we apply the theorems 27.2 and 27.3. From the theorems 27.2 we derive the following equalities: c (a + b) - c (a) - c (b) = c ( a) - c (a) = (a + b, c) - (a, c) - (b, c) c, |c|2 ( a, c) - (a, c) c. |c|2

Due to the theorem 27.3 the mapping c is a linear mapping (see Definition 27.4). Therefore the left hand sides of the ab ove equalities are zero. Now due to c = 0 we conclude that the


§ 29. . . . IN A SKEW-ANGULAR BASIS.

75

numerators of the fractions in their right hand sides are also zero. This fact proves the prop erties 2) and 3) from the theorem 28.1 are valid in the case c = 0. According to the definition 26.1 the scalar product (a, a) is equal to zero for a = 0. Otherwise, if a = 0, the formula (26.1) is applied where we should set b = a. This yields = 0 and (a, a) = |a|2 > 0. This inequality proves the prop erty 4) and completes the proof of the theorem 28.1 in whole. Theorem 28.2. Apart from the prop erties 1)­4), the scalar product of vectors p ossesses the following two prop erties fulfilled for any three vectors a, b, c and for any numb er : 5) (c, a + b) = (c, a) + (a, b); 6) (c, a) = (c, a). Definition 28.2. The prop erties 5) and 6) in the theorem 28.2 are called the prop erties of linearity with respect to the second multiplicand. The prop erties 5) and 6) are easily derived from the prop erties 2) and 3) by applying the prop erty 1). Indeed, we have (c, a + b) = (a + b, c) = (a, c) + (b, c) = (c, a) + (c, b), (c, a) = ( a, c) = (a, c) = (c, a). These calculations prove the theorem 28.2. § 29. Calculation of the scalar product through the coordinates of vectors in a skew-angular basis. Let e1 , e2 , e3 b e some arbitrary basis in the space E. According to the definition 18.1, this is an ordered triple of non-coplanar vectors. The arbitrariness of a basis means that no auxiliary restrictions are imp osed onto the vectors e1 , e2 , e3 , except for


76

CHAPTER I. VECTOR ALGEBRA.

non-coplanarity. In particular, this means that the angles b etween the vectors e1 , e2 , e3 in an arbitrary basis should not b e right angles. For this reason such a basis is called a skew-angular basis and abbreviated as SAB. Definition 29.1. In this b ook a skew-angular basis (SAB) is understood as an arbitrary basis. Thus, let e1 , e2 , e3 b e some skew-angular basis in the space E and let a and b b e two free vectors given by its coordinates in this basis. We write this fact as follows: a= a1 a2 , a3 b b= b b
1 2 3

(29.1)

Unlike (21.1), instead of the arrow sign in (29.1) we use the equality sign. Doing this, we emphasize the fact that once a basis is fixed, vectors are uniquely identified with their coordinates. The conditional writing (29.1) means that the vectors a and b are presented by the following expansions:
3 3

a=
i =1

a ei ,

i

b=
j =1

b j ej .

(29.2)

Substituting (29.2) into the scalar product (a, b), we get
3 3

(a, b) =
i =1

a ei ,
j =1

i

bj e

j

.

(29.3)

In order to transform the formu 2) and 5) of the scalar product Due to these prop erties we can and j out of the brackets of the
3 3

las (29.3) from the take the scalar pr

we apply the prop erties theorems 28.1 and 28.2. summation signs over i oduct: (29.4)

(a, b) =
i =1 j =1

(ai ei , b j ej ).


§ 29. . . . IN A SKEW-ANGULAR BASIS.

77

Then we apply the prop erties 3) and 6) from the theorems 28.1 and 28.2. Due to these prop erties we can take the numeric factors ai and b j out of the brackets of the scalar product in (29.4):
3 3

(a, b) =
i =1 j =1

ai b j (ei , ej ).

(29.5)

The quantities (ei , ej ) in the formula (29.5) dep end on a basis e1 , e2 , e3 , namely on the lengths of the basis vectors and on the angles b etween them. They do not dep end on the vectors a and b. The quantities (ei , ej ) constitute an array of nine numb ers gij = (ei , ej ) (29.6)

enumerated by two lower indices. The comp onents of the array (29.6) are usually arranged into a square matrix: g11 G = g21 g31 g12 g22 g32 g13 g23 g33 (29.7)

Definition 29.2. The matrix (29.7) with the comp onents (29.6) is called the Gram matrix of a basis e1 , e2 , e3 . Taking into account the notations (29.6), we write the formula (29.5) in the following way:
3 3

(a, b) =
i =1 j =1

ai b j gij .

(29.8)

Definition 29.3. The formula (29.8) is called the formula for calculating the scalar product through the coordinates of vectors in a skew-angular basis.
CopyRight c Sharipov R.A., 2010.


78

CHAPTER I. VECTOR ALGEBRA.

The formula (29.8) can b e written in the matrix form g11 (a, b) = a1 a2 a3 · g21 g31 g12 g22 g32 g13 g23 g33 b ·b b
1 2 3

(29.9)

Note that the coordinate column of the vector b in the formula (29.9) is used as it is, while the coordinate column of the vector a is transformed into a row. Such a transformation is known as matrix transposing (see [7]). Definition 29.4. A transformation of a rectangular matrix under which the element in the intersection of i-th row and j -th column is taken to the intersection of j -th row and i-th column is called the matrix transposing. It is denoted by means of the sign . In the TEX and LaTEX computer packages this sign is coded by the op erator \top. The op eration of matrix transp osing can b e understood as the mirror reflection with resp ect to the main diagonal of a matrix Taking into account the notations (29.1), (29.7) , and the definition 29.4, we can write the matrix formula (29.9) as follows: (a, b) = a · G · b. (29.10)

In the right hand side of the formula (29.10) the vectors a and b are presented by their coordinate columns, while the transformation of one of them into a row is written through the matrix transp osing. Exercise 29.1. Show that for an arbitrary rectangular matrix A the equality (A ) = A is fulfilled. Exercise 29.2. Show that for the product of two matrices A and B the equality (A · B ) = B · A is fulfilled.


§ 30. SYMMETRY OF THE GRAM MATRIX.

79

Exercise 29.3. Define the Gram matrices for bases on a line and for bases on a plane. Write analogs of the formulas (29.8), (29.9), and (29.10) for the scalar product of vectors lying on a line and on a plane. § 30. Symmetry of the Gram matrix. Definition 30.1. A square matrix A is called symmetric, if it is preserved under transp osing, i. e. if the following equality is fulfilled: A = A. Gram matrices p ossesses many imp ortant prop erties. One of these prop erties is their symmetry. Theorem 30.1. The Gram matrix G of any basis e1 , e2 , e in the space E is symmetric.
3

Proof. According to the definition 30.1 the symmetry of G is expressed by the formula G = G. According to the definition 29.4, the equality G = G is equivalent to the relationship gij = gj
i

(30.1)

for the comp onents of the matrix G. As for the relationship (30.1), up on applying (29.6), it reduces to the equality (ei , ej ) = (ej , ei ) which is fulfilled due to the symmetry of the scalar product (see Theorem 28.1 and Definition 28.1). Note that the coordinate columns of the vectors a and b enter the right hand side of the formula (29.10) in somewhat unequal way -- one of them is transp osed, the other is not transp osed. The symmetry of the matrix G eliminates this difference. Redesignating the indices i and j in the double sum


80

CHAPTER I. VECTOR ALGEBRA.

(29.8) and taking into account the relationship (30.1) for the comp onents of the Gram matrix, we get
3 3 3 3

(a, b) =
j =1 i =1

aj b i gj i =
i =1 j =1

b i aj gij .

(30.2)

In the matrix form the formula (30.2) is written as follows: (a, b) = b · G · a. (30.3)

The formula (30.3) is analogous to the formula (29.10) , but in this formula the coordinate column of the vector b is transp osed, while the coordinate column of the vector a is not transp osed. Exercise 30.1. Formulate and prove a theorem analogous to the theorem 30.1 for bases on a plane. Is it necessary to formulate such a theorem for bases on a line. § 31. Orthonormal basis. Definition 31.1. A basis on a straight line consisting of a nonzero vector e is called an orthonormal basis, if e is a unit vector, i. e. if |e| = 1. Definition 31.2. A basis on a plane, consisting of two noncollinear vectors e1 , e2 , is called an orthonormal basis, if the vectors e1 and e2 are two vectors of the unit lengths p erp endicular to each other. Definition 31.3. non-coplanar vectors if the vectors e1 , e2 , p erp endicular to each A basis in the space E consisting of three e1 , e2 , e3 is called an orthonormal basis e3 are three vectors of the unit lengths other.

In order to denote an orthonormal basis in each of the there cases listed ab ove we use the abbreviation ONB. According to


§ 32. . . . OF AN ORTHONORMAL BASIS.

81

the definition 29.1 the orthonormal basis is not opp osed to a skew-angular basis SAB, it is a sp ecial case of such a basis. Note that the unit lengths of the basis vectors of an orthonormal basis in the definitions 31.1, 31.2, and 31.2 mean that their lengths are not one centimeter, not one meter, not one kilometer, but the pure numeric unity. For this reason all geometric realizations of such vectors are conditionally geometric (see § 2). Like velocity vectors, acceleration vectors, and many other physical quantities, basis vectors of an orthonormal basis can b e drawn only up on choosing some scaling factor. Such a factor in this particular case is needed for to transform the numeric unity into a unit of length. § 32. Gram matrix of an orthonormal basis. Let e1 , e2 , e3 b e some orthonormal basis in the space E. According to the definition 31.3 the vectors e1 , e2 , e3 satisfy the following relationships: |e1 | = 1, e1 e2 , |e2 | = 1, e2 e3 , |e3 | = 1, e3 e1 . (32.1)

Applying (32.1) and (29.6), we find the comp onents of the Gram matrix for the orthonormal basis e1 , e2 , e3 : gij = 1 0 for i = j, for i = j. (32.2)

From (32.2) we immediately derive the following theorem. Theorem 32.1. The Gram matrix (29.7) of any orthonormal basis is a unit matrix: G= 100 010 001 = 1. (32.3)


82

CHAPTER I. VECTOR ALGEBRA.

Let's recall that a numeric array Definition 23.1). equality (32.2) can

the comp onents of a unit matrix constitute which is called the Kronecker symb ol (see Therefore, taking into account (23.3), the b e written as: gij = ij . (32.4)

The Kronecker symb ol in (32.4) inherits the lower p osition of indices from gij . Therefore it is different from the Kronecker symb ol in (23.4) . Despite b eing absolutely identical, the comp onents of the unit matrix in (32.3) and in (23.1) are of absolutely different nature. Having negotiated to use indices on upp er and lower levels (see Definition 20.1), now we are able to reflect this difference in denoting these comp onents. § 33. Calculation of the scalar product through the coordinates of vectors in an orthonormal basis. According to the definition 29.1 the term skew-angular basis is used as a synonym of an arbitrary basis. For this reason an orthonormal basis is a sp ecial case of a skew-angular basis and we can use the formula (29.8) , taking into account (32.4):
3 3

(a, b) =
i =1 j =1

ai b i ij .

(33.1)

In calculating the sum over j in (33.1), it is the inner sum here, the index j runs over three values and only for one of these three values, where j = i, the Kronecker symb ol ij is nonzero. For this reason wee can retain only one summand of the inner sum over j in (33.1) , omitting other two summands:
3

(a, b) =
i =1

ai b i ii .

(33.2)


§ 34. RIGHT AND LEFT TRIPLES OF VECTORS.

83

We know that ii = 1. Therefore the formula (33.2) turns to
3

(a, b) =
i =1

ai b i .

(33.3)

Definition 33.1. The formula (33.3) is called the formula for calculating the scalar product through the coordinates of vectors in an orthonormal basis. Note that the sums in the formula (29.8) satisfy the index setting rule from the definition 24.8, while the sum in the formula (33.3) breaks this rule. In this formula the summation index has two entries and b oth of them are in the upp er p ositions. This is a p eculiarity of an orthonormal basis. It is more symmetric as compared to a general skew-angular basis and this symmetry hides some rules that reveal in a general non-symmetric bases. The formula (33.3) has the following matrix form:
1 2 3

(a, b) = a

a

a

b ·b b

1 2 3

.

(33.4)

Taking into account the notations (29.1) and taking into account the definition 29.4, the formula (33.4) can b e abbreviated to (a, b) = a · b. (33.5)

The formula (33.4) can b e derived from the formula (29.9) , while the formula (33.5) can b e derived from the formula (29.10) . § 34. Right and left triples of vectors. The concept of orientation. Definition 34.1. An ordered triple of vectors is a list of three vectors for which the order of listing vectors is fixed.


84

CHAPTER I. VECTOR ALGEBRA.

Definition 34.2. An ordered triple of non-coplanar vectors a1 , a2 , a3 is called a right triple if, when observing from the end of the third vector, the shortest rotation from the first vector toward the second vector is seen as a counterclockwise rotation. In the definition 34.2 we implicitly assume that the geometric realizations of the vectors a1 , a2 , a3 with some common initial p oint are considered as it is shown in Fig. 34.1. Def a1 , a2 , of the toward inition 34.3. An a3 is called a left third vector, the the second vector ordered triple of non-coplanar vectors triple if, when observing from the end shortest rotation from the first vector is seen as a clockwise rotation.

A given rotation ab out a given axis when observing from a given p osition could b e either a clockwise rotation or a counterclockwise rotation. No other options are available. For this reason each ordered triple of non-coplanar vectors is either left or right. No other triples sorted by this criterion are available. Definition 34.4. The prop erty of ordered triples of noncoplanar vectors to b e left or right is called their orientation. § 35. Vector product. Let a and b b e two non-collinear free vectors. Let's lay their - - - - geometric realizations a = OA and b = OB at some arbitrary p oint O. In this case the vectors a and b define a plane AOB and lie on this plane. The angle b etween the vectors a and b is determined according to Fig. 26.1. Due to a b this angle ranges in the interval 0 < < and hence sin = 0. Let's draw a line through the p oint O p erp endicular to the plane AO B and denote this line through c. The line c is
CopyRight c Sharipov R.A., 2010.


§ 35. VECTOR PRODUCT.

85

- - - - p erp endicular to the vectors a = OA and b = OB : c a, c b. (35.1)

It is clear that the conditions (35.1) fix a unique line c passing through the p oint O (see Theorems 1.1 and 1.3 in Chapter IV of the b ook [6]). There are two directions on the line c. In Fig~5.1 they are 3 given by the vectors c and ~. c The vectors a, b, c constitute a right triple, while a, b, ~ is a left c triple. So, sp ecifying the orientation chooses one of two p ossible directions on the line c. Definition 35.1. The vector product of two non-collinear vectors a and b is a vector c = [a, b] which is determined by the following three conditions: 1) c a and c b; 2) the vectors a, b, c form a right triple; 3) |c| = |a| |b| sin . In the case of collinear vectors a and b their vector product [a, b] is taken to b e zero by definition. A comma is the multiplication sign in the writing the vector product, not by itself, but together with square brackets surrounding the whole expression. These brackets are natural delimiters for multiplicands: the first multiplicand is an expression b etween the op ening bracket and the comma, while the second multiplicand is an expression b etween the comma and the closing bracket. Therefore in complicated expressions no auxiliary delimiters are required. For example, in the formula [a + b, c + d]


86

CHAPTER I. VECTOR ALGEBRA.

the sums a + b and c + d are calculated first, then the vector multiplication is p erformed. A remark. Often the vector product is written as a â b. Even the sp ecial term «cross product» is used. However, to my mind, this notation is not good. It is misleading since the cross sign is sometimes used for denoting the product of numb ers when a large formula is split into several lines. A remark. The physical nature of the vector product [a, b] often differs from the nature of its multiplicands a and b. Even if the lengths of the vectors a and b are measured in length units, the length of their product [a, b] is measured in units of area. Exercise 35.1. Show that the vector product c = [a, b] of two free vectors a and b is a free vector and, b eing a free vector, it does not dep end on where the p oint O in Fig. 35.1 is placed. § 36. Orthogonal pro jection onto a plane. Let a = 0 b e some nonzero free vector. According to the theorem 27.1, each free vector b has the expansion b = b + b (36.1)

relative to the vector a, where the vector b is collinear to the vector a, while the vector b is p erp endicular to the vector a. Recall that through a we denoted a mapping that associates each vector b with its comp onent b in the expansion (36.1). Such a mapping was called the orthogonal pro jection onto the direction of the vector a = 0 (see Definition 27.3). Definition 36.1. The mapping a that associates each free vector b with its p erp endicular comp onent b in the expansion (36.1) is called the orthogonal projection onto a plane perpendicular to the vector a = 0 or, more exactly, the orthogonal projection onto the orthogonal complement of the vector a = 0. Definition 36.2. The orthogonal complement of a free vector


§ 36. ORTHOGONAL PROJECTION ONTO A PLANE.

87

a is the collection of all free vectors x p erp endicular to a: = {x : x a}. (36.2)

The orthogonal complement (36.2) of a nonzero vector a = 0 can b e visualized as a plane if we choose one of its geometric - - realizations a = OA . Indeed, let's lay various vectors p erp endicular to a at the p oint O. The ending p oints of such vectors fill the plane shown in Fig. 36.1. The prop erties of the orthogonal pro jections onto a line a from the definition 27.1 and the orthogonal pro jections onto a plane a from the definition 36.1 are very similar. Indeed, we have the following theorem. Theorem 36.1. For any nonzero vector a = 0 the orthogonal pro jection a onto a plane p erp endicular to the vector a is a linear mapping. Proof. In order to prove the theorem 36.1 we write the relationship (36.1) as follows: b = a (b) + a (b). (36.3)

The relationship (36.3) is an identity, it is fulfilled for any vector b. First we replace the vector b by b + c in (36.3), then we replace b by b in (36.3). As a result we get two relationships a (b + c) = b + c - a (b + c), (36.4) (36.5)

a ( b) = b - a ( b).

Due to the theorem 27.3 the mapping a is a linear mapping. For this reason the relationships (36.4) and (36.5) can b e


88

CHAPTER I. VECTOR ALGEBRA.

transformed into the following two relationships: a (b + c) = b - a (b) + c - a (c), (36.6) (36.7)

a ( b) = (b - a (b)).

The rest is to apply the identity (36.3) to the relationships (36.6) and (36.7) . As a result we get a (b + c) = a (b) + a (c), a ( b) = a (b). (36.8) (36.9)

The relationships (36.8) and (36.9) are exactly the linearity conditions from the definition 27.4 written for the mapping a . The theorem 36.1 is proved. § 37. Rotation ab out an axis. Let a = 0 b e some nonzero free vector and let b b e some arbi- - trary free vector. Let's lay the vector b = B O at some arbitrary - - p oint B . Then we lay the vector a = OA at the terminal p oint of - - - - the vector B O . The vector a = OA is nonzero. For this reason it defines a line OA. We take this line for the rotation axis. Let's denote through a the rotation of the space E ab out the axis OA by the angle (see Fig. 37.1). - - The vector a = OA fixes one of two directions on the rotation axis. At the same tame this vector fixes the p ositive direction of rotation ab out the axis OA. Definition 37.1. The rotation ab out an axis OA with with - - fixed direction a = OA on it is called a p ositive rotation if, b eing observed from the termi- - nal p oint of the vector OA , i. e.


§ 37. ROTATION ABOUT AN AXIS.

89

when looking from the p oint A toward the p oint O, it occurs in the counterclockwise direction. Taking into account the definition 37.1, we can consider the rotation angle as a signed quantity. If > 0, the rotation a occurs in the p ositive direction with resp ect to the vector a, if < 0, it occurs in the negative direction. - - Let's apply the rotation mapping a to the vectors a = OA - - and b = B O in Fig. 37.1. The p oints A and O are on the rotation axis. For this reason under the rotation a the p oints - - A and O stay at their places and the vector a = OA does not - - change. As for the vector b = B O , it is mapp ed onto another - - ~ vector B O . Now, applying parallel translations, we can replicate - - - - ~ ~ the vector B O up to a free vector ~ = B O (see Definitions 4.1 b and 4.2). The vector ~ is said to b e produced from the vector b b by applying the mapping a and is written as ~ = (b). b a (37.1)

Lemma 37.1. The free vector ~ = a (b) in (37.1) produced b from a free vector b by means of the rotation mapping a does not dep end on the choice of a geometric realization of the vector a defining the rotation axis and on a geometric realization of the vector b itself. Definition 37.2. The mapping a acting up on free vectors of the space E and taking them to other free vectors in E is called the rotation by the angle ab out the vector a.

Exercise 37.1. Rotations and parallel translations b elong to the class of mappings preserving lengths of segments and measures of angles. They take each segment to a congruent segment and each angle to a congruent angle (see [6]). Let p b e some parallel translation, let p-1 b e its inverse parallel translation, and let b e a rotation by some angle ab out some axis. Prove that the


90

CHAPTER I. VECTOR ALGEBRA.

~ comp osite mapping = p p-1 is the rotation by the same angle ab out the axis produced from the axis of by applying the parallel translation p to it. Exercise 37.2. Apply the result of the exercise 37.1 for proving the lemma 37.1. Theorem 37.1. For any nonzero free vector a = 0 and for any angle the rotation a by the angle ab out the vector a is a linear mapping of free vectors. Proof. In order to prove the theorem we need to insp ect the conditions 1) and 2) from the definition 27.4 for the mapping a . Let's b egin with the first of these conditions. Assume that b and c are two free vectors. Let's build their geometric realizations - - - - - - b = B C and c = C O . Then the vector B O is the geometric realization for the sum of vectors b + c. Now let's choose a geometric realization for the vector a. It determines the rotation axis. According to the lemma 37.1 the actual place of such a geometric realization does not matter for the ultimate definition of the mapping a as applied to free - - vectors. But for the sake of certainty we choose a = OA . Let's apply the rotation by the angle ab out the axis OA to the p oints B , C , and O. The p oint O is on the rotation axis. Therefore it is not moved. The p oints B and C are moved to the ~ ~ p oints B and C resp ectively, while the triangle B C O is moved to ~ C O. As a result we get the following relationships: the triangle B ~
a a a

- - - - ~~ BC = BC , - - - - ~ CO = CO , - - - - ~ BO = BO .

(37.2)

- - ~~ But the vectors B C and for the vectors ~ = (b) b
a

- - ~ C O in (37.2) are geometric realizations - - ~ ~ and c = a (c), while B O is a geometric


§ 38. THE RELATION OF THE VECTOR PRODUCT . . .
realization for the vector a (b + c). Hence we have

91

- - - - - - ~ ~~ ~ a (b + c) = B O = B C + C O = a (b) + a (c).

(37.3)

The chain of equalities (37.3) proves the first linearity condition from the definition 27.4 as applied to a . Let's proceed to proving the second linearity condition. It is more simple than the first one. Multiplying a vector b by a numb er , we make the length of its geometric realizations || times as greater. If > 0, geometric realizations of the vector b are codirected to geometric realizations of b. If < 0, they are opp osite to geometric realizations of b. And if = 0, geometric realizations of the vector b do vanish. Let's apply the rotation by the angle ab out some geometric realization of the vector a = 0 some geometric realizations of the vectors b and b. Such a mapping preserves the lengths of vectors. Hence it preserves all the relations of their lengths. Moreover it maps straight lines to straight lines and preserves the order of p oints on that straight lines. Hence codirected vectors are mapp ed to codirected ones and opp osite vectors to opp osite ones resp ectively. As a result
a ( b) = a (b).

(37.4)

The relationship (37.4) completes the proof of the linearity for the mapping a as applied to free vectors. § 38. The relation of the vector product with pro jections and rotations. Let's consider two non-collinear vectors a b and their vector product c = [a, b]. The length of the vector c is determined by the lengths of a and b and by the angle b etween them: |c| = |a| |b| sin .
CopyRight c Sharipov R.A., 2010.

(38.1)


92

CHAPTER I. VECTOR ALGEBRA.

The vector c lies on the plane p erp endicular to the vector b (see Fig. 38.1). Let's denote through a the orthogonal projection of the vector a onto the plane , i e. we set a = b (a). (38.2)

The length of the ab ove vector (38.2) is determined by the formula |a | = |a| sin . Comparing this formula with (38.1) and taking into account Fig. 38.1, we conclude that in order to sup erp ose the vector a with the vector c one should first rotate it counterclockwise by the right angle ab out the vector b and then multiply by the negative numb er -|b|. This yields the formula [a, b] = -|b| ·
/2 b

b (a) .

(38.3)

The formula (38.3) sets the relation of the vector product with the two mappings b and b/2 . One of them is the pro jection onto the orthogonal complement of the vector b, while the other is the rotation by the angle /2 ab out the vector b. The formula (38.3) is applicable provided the vector b is nonzero: b = 0, (38.4)

while the condition a b can b e broken. If a b b oth sides of the formula (38.4) vanish, but the formula itself remains valid. § 39. Prop erties of the vector product. Theorem 39.1. The vector product of vectors p ossesses the following four prop erties fulfilled for any three vectors a, b, c and


§ 39. PROPERTIES OF THE VECTOR PRODUCT.

93

for any numb er : 1) [a, b] = -[b, a]; 2) [a + b, c] = [a, c] + [b, c]; 3) [ a, c] = [a, c]; 4) [a, b] = 0 if and only if the vectors a and b are collinear, i. e. if a b. Definition 39.1. The prop erty 1) in the theorem 39.1 is called anticommutativity; the prop erties 2) and 3) are called the prop erties of linearity with respect to the first multiplicand; the prop erty 4) is called the vanishing condition. Proof of the theorem 39.1. The prop erty of anticommutativity 1) is derived immediately from the definition 35.1. Let a b. Exchanging the vectors a and b, we do not violate the first and the third conditions for the triple of vectors a, b, c in the definition 35.1, provided they are initially fulfilled. As for the direction of rotation in Fig. 35.1, it changes for the opp osite one. Therefore, if the triple a, b, c is right, the triple b, a, c is left. In order to get a right triple the vectors b and a should b e complemented with the vector -c. This yield the equality [a, b] = -[b, a] (39.1)

for the case a b. If a b, b oth sides of the equality (39.1) do vanish. So the equality remains valid in this case too. Let c = 0. The prop erties of linearity 1) and 2) for this case are derived with the use of the formula (38.3) and the theorems 36.1 and 37.1. Let's write the formula (38.3) as [a, c] = -|c| ·
/2 c

c (a) .

(39.2)

Then we change the vector a in (39.2) for the sum of vectors a + b and apply the theorems 36.1 and 37.1. This yields [a + b, c] = -|c| ·
/2 c

c (a + b) = -|c| ·


94

CHAPTER I. VECTOR ALGEBRA.

·

/2 c

- |c| ·

c (a) + c (b) = -|c| ·
/2 c

/2 c

c (b) = [a, c] + [b, c].

c (a) -

Now we change the vector a in (39.2) for the product a and then apply the theorems 36.1 and 37.1 again: [ a, c] = -|c| ·
/2 c

= - |c| ·

c ( a) = -|c| ·
/2 c

/2 c

c (a) =

c (a) = [a, c].

The calculations which are p erformed ab ove prove the equalities 2) and 3) in the theorem 39.1 for the case c = 0. If c = 0, b oth sides of these equalities do vanish and they app ear to b e trivially fulfilled. Let's proceed to proving the fourth item in the theorem 39.1. For a b the vector product [a, b] vanishes by the definition 35.1. Let a b. In this case b oth vectors a and b are nonzero, while the angle b etween them differs from 0 and . For this reason sin = 0. Summarizing these restrictions and applying the item 3) of the definition 35.1, we get |[a, b]| = |a| |b| sin = 0, i. e. for a b the vector product [a, b] cannot vanish. The proof of the theorem 39.1 is over. Theorem 39.2. Apart from the prop erties 1)­4), the vector product p ossesses the following two prop erties which are fulfilled for any vectors a, b, c and for any numb er : 5) [c, a + b] = [c, a] + [a, b]; 6) [c, a] = [c, a]. Definition 39.2. The prop erties 5) and 6) in the theorem 39.2 are called the prop erties of linearity with respect to the second multiplicand.


§ 40. STRUCTURAL CONSTANTS . . .

95

The prop erties 5) and 6) are easily derived from the prop erties 2) and 3) by applying the prop erty 1). Indeed, we have [c, a + b] = -[a + b, c] = -[a, c] - [b, c] = [c, a] + [c, b], [c, a] = -[ a, c] = - [a, c] = [c, a]. These calculations prove the theorem 39.2. § 40. Structural constants of the vector product. Let e1 , e2 , e3 take two vectors product [ei , ej ]. e1 , e2 , e3 . Such b e some arbitrary basis in the space E. Let's ei and ej of this basis and consider their vector The vector [ei , ej ] can b e expanded in the basis an expansion is usually written as follows: (40.1)

1 2 3 [ei , ej ] = Cij e1 + Cij e2 + Cij e3 .

2 1 The expansion (40.1) contains three coefficients Cij , Cij and 3 Cij . However, the indices i and j in it run indep endently over three values 1, 2, 3. For this reason, actually, the formula (40.1) represent nine expansions, the total numb er of coefficients in it is equal to twenty seven. The formula (40.1) can b e abbreviated in the following way: 3

[ei , ej ] =
k =1

k Ci j ek .

(40.2)

Let's apply the theorem 19.1 on the uniqueness of the expansion of a vector in a basis to the expansions of [ei , ej ] in (40.1) or in (40.2). As a result we can formulate the following theorem. Theorem 40.1. Each basis e1 , e2 , e3 in the space E is assok ciated with a collection of twenty seven constants Cij which are determined uniquely by this basis through the expansions (40.2).
k Definition 40.1. The constants Cij , which are uniquely de-


96

CHAPTER I. VECTOR ALGEBRA.

termined by a basis e1 , e2 , e3 through the expansions (40.2), are called the structural constants of the vector product in this basis. The structural constants of the vector product are similar to the comp onents of the Gram matrix for a basis e1 , e2 , e3 (see Definition 29.2). But they are more numerous and form a three index array with two lower indices and one upp er index. For this reason they cannot b e placed into a matrix. § 41. Calculation of the vector product through the coordinates of vectors in a skew-angular basis. Let e1 , e2 , e3 b e some skew-angular basis. According to the definition 29.1 the term skew-angular basis in this b ook is used as a synonym of an arbitrary basis. Let's choose some arbitrary vectors a and b in the space E and consider their expansions
3 3

a=
i =1

a ei ,

i

b=
j =1

bj e

j

(41.1)

in the basis e1 , e2 , e3 . Substituting (41.1) into the vector product [a, b], we get the following formula:
3 3

[a, b] =
i =1

a ei ,
j =1

i

b j ej .

(41.2)

In order to transform the formula (41.2) we apply the prop erties 2) and 5) of the vector product (see Theorems 39.1 and 39.2). Due to these prop erties we can bring the summation signs over i and j outside the brackets of the vector product:
3 3

[a, b] =
i =1 j =1

[ai ei , b j ej ].

(41.3)

Now let's apply the prop erties 3) and 6) from the theorems 39.1 and 39.2. Due to these prop erties we can bring the numeric


§ 42. STRUCTURAL CONSTANTS . . .

97

factors ai and b j outside the brackets of the vector product (41.3):
3 3

[a, b] =
i =1 j =1

ai b j [ei , ej ].

(41.4)

The vector products [ei , ej ] in the formula (41.4) can b e replaced by their expansions (40.2). Up on substituting (40.2) into (41.4) the formula (41.4) is written as follows:
3 3 3 k ai b j C i j e k . i =1 j =1 k =1

[a, b] =

(41.5)

Definition 41.1. The formula (41.5) is called the formula for calculating the vector product through the coordinates of vectors in a skew-angular basis. § 42. Structural constants of the vector product in an orthonormal basis. Let's recall that an orthonormal basis (ONB) in the space E is a basis comp osed by three unit vectors p erp endicular to each other (see Definition 31.3). By their orientation, triples of noncoplanar vectors in the space E are sub divided into right and left triples (see Definition 34.4). Therefore all bases in the space E are sub divided into right bases and left bases, which applies to orthonormal bases as well. Let's consider some right orthonormal basis e1 , e2 , e3 . It is shown in Fig. 42.1. Using the definition 35.1, one can calculate various pairwise vector products of the vectors comp osing this basis. Since the geometry of a right


98

CHAPTER I. VECTOR ALGEBRA.

ONB is rather simple, we can p erform these calculations up to an explicit result and comp ose the multiplication table for e1 , e2 , e3 : [e1 , e1 ] = 0, [e2 , e1 ] = -e3 , [e3 , e1 ] = e2 , [e1 , e2 ] = e3 , [e2 , e2 ] = 0, [e3 , e2 ] = -e1 , [e1 , e3 ] = -e2 , [e3 , e3 ] = 0.

[e2 , e3 ] = e1 ,

(42.1)

Let's choose the first of the relationships (42.1) and write its right hand side in the form of an expansion in the basis e1 , e2 , e3 : [e1 , e1 ] = 0 e1 + 0 e2 + 0 e3 . (42.2)

Let's compare the expansion (42.2) with the expansion (40.1) written for the case i = 1 and j = 1: [e1 , e1 ] = C
1 11

e1 + C

2 11

e2 + C

3 11

e3 .

(42.3)

Due to the uniqueness of the expansion of a vector in a basis (see Theorem 19.1) from (42.2) and (42.3) we derive C
1 11

= 0,

C

2 11

= 0,

C

3 11

= 0.

(42.4)

Now let's choose the second relationship (42.1) and write its right hand side in the form of an expansion in the basis e1 , e2 , e3 : [e1 , e2 ] = 0 e1 + 0 e2 + 1 e3 . (42.5)

Comparing (42.5) with the expansion (40.1) written for the case i = 1 and j = 2, we get the values of the following constants: C
1 12

= 0,

C

2 12

= 0,

C

3 12

= 1.

(42.6)

Rep eating this procedure for all relationships (42.1), we can get the complete set of relationships similar to (42.4) and (42.6).
CopyRight c Sharipov R.A., 2010.


§ 43. LEVI-CIVITA SYMBOL.

99

Then we can organize them into a single list: C C C C C C C C C
1 11 1 12 1 13 1 21 1 22 1 23 1 31 1 32 1 33

= 0, = 0, = 0, = 0, = 0, = 1, = 0, = 0, = -1,

C C C C C C C C C

2 11 2 12 2 13 2 21 2 22 2 23 2 31 2 32 2 33

= 0, = 0, = 0, = 0, = 0, = 1, = 0, = 0, = -1,

C C C C C C C C C

3 11 3 12 3 13 3 21 3 22 3 23 3 31 3 32 3 33

= 0, = 1, = 0, = 0, = 0, = 0, = 1, = 0. = -1, (42.7)

The formulas (42.7) determine all of the 27 structural constants of the vector product in a right orthonormal basis. Let's write this result as a theorem. Theorem 42.1. For any right orthonormal basis e1 , e2 , e3 in the space E the structural constants of the vector product are determined by the formulas (42.7) . Theorem 42.2. For any left orthonormal basis e1 , e2 , e3 in the space E the structural constants of the vector product are derived from (42.7) by changing signs «+» for «-» and vise versa. Exercise 42.1. Draw a left orthonormal basis and, applying the definition 35.1 to the pairwise vector products of the basis vectors, derive the relationships analogous to (42.1) . Then prove the theorem 42.2. § 43. Levi-Civita symb ol. Let's examine the formulas (42.7) for the structural constants of the vector product in a right orthonormal basis. One can


100

CHAPTER I. VECTOR ALGEBRA.

easily observe the following pattern in them:
k Cij = 0 if there are coinciding values of the indices i, j, k.

(43.1)

The condition (43.1) describ es all of the cases where the struck tural constants in (42.7) do vanish. The cases where Cij = 1 are describ ed by the condition
k Cij = 1 if the indices i, j, k take the values (1, 2, 3), (2, 3, 1), or (3, 1, 2).

(43.2)

k Finally, the cases where Cij = -1 are describ ed by the condition k Cij = -1 if the indices i, j, k take the values (1, 3, 2), (3, 2, 1), or (2, 1, 3).

(43.3)

The triples of numb ers in (43.2) and (43.3) constitute the complete set of various p ermutations of the numb ers 1, 2, 3: (1, 2, 3), (1, 3, 2), (2, 3, 1), (3, 2, 1), (3, 1, 2), (2, 1, 3). (43.4)

The first three p ermutations in (43.4) are called even permutations. They are produced from the right order of the numb ers 1, 2, 3 by applying an even numb er of pairwise transp ositions to them. Indeed, we have (1, 2, 3); (1, 2, 3) - (2, 1, 3) - (2, 3, 1); (1, 2, 3) - (1, 3, 2) - (3, 1, 2). The rest three p ermutations in (43.4) are called odd permutations. In the case of these three p ermutations we have (1, 2, 3) - (1, 3, 2);
1 1 2 1 2


§ 43. LEVI-CIVITA SYMBOL.

101

(1, 2, 3) - (3, 2, 1); (1, 2, 3) - (2, 1, 3). Definition 43.1. The p ermutations (43.4) constitute a set, which is usually denoted through S3 . If S3 , then (-1) means the parity of the p ermutation : (-1) = 1 if the p ermutation is even; -1 if the p ermutation is odd.
1

1

Zeros, unities, and minus unities from (43.1), (43.2), and (43.3) are usually united into a single numeric array: 0 if there are coinciding values of the indices i, j, k ; 1 if the values of the indices i, j, k form an even p ermutation of ij k = ij k = (43.5) the numb ers 1, 2, 3; -1 if the values of the indices i, j, k form an odd p ermutation of the numb ers 1, 2, 3.

Definition 43.2. The numeric array determined by the formula (43.5) is called the Levi-Civita symb ol.

When writing the comp onents of the Levi-Civita symb ol either three upp er indices or three lower indices are used. Thus we emphasize the equity of all these three indices. Placing indices on different levels in the Levi-Civita symb ol is nor welcome. Summarizing what was said ab ove, the formulas (43.1), (43.2), and (43.3) are written as follows:
k Cij = ij k .

(43.6)

Theorem 43.1. For any right orthonormal basis e1 , e2 , e3 the


102

CHAPTER I. VECTOR ALGEBRA.

structural constants of the vector product in such a basis are determined by the equality (43.6). In the case of a left orthonormal basis we have the theorem 42.2. It yields the equality
k Cij = -ij k .

(43.7)

Theorem 43.2. For any left orthonormal basis e1 , e2 , e3 the structural constants of the vector product in such a basis are determined by the equality (43.7). Note that the equalities (43.6) and (43.7) violate the index setting rule given in the definition 24.9. The matter is that the k structural constants of the vector product Cij are the comp onents of a geometric ob ject. The places of their indices are determined by the index setting convention, which is known as Einstein's tensorial notation (see Definition 20.1). As for the Levi-Civita symb ol, it is an array of purely algebraic origin. The most imp ortant prop erty of the Levi-Civita symb ol is its complete skew symmetry or complete antisymmetry. It is expressed by the following equalities: i i
jk jk

= -j ik , = -j ik ,

i i

jk jk

= -ikj , = -ikj ,

i i

jk jk

= -kj i , = -kj i .

(43.8)

The equalities (43.8) mean, that under the transp osition of any two indices the quantity ij k = ij k changes its sign. These equalities are easily derived from (43.5). § 44. Calculation of the vector product through the coordinates of vectors in an orthonormal basis. Let's recall that the term skew-angular basis in this b ook is used as a synonym of an arbitrary basis (see Definition 29.1). Let e1 , e2 , e3 b e a right orthonormal basis. It can b e treated as a


§ 44. . . . IN AN ORTHONORMAL BASIS.

103

sp ecial case of a skew-angular basis. Substituting (43.6) into the formula (41.5), we obtain the formula
3 3 3

[a, b] =
i =1 j =1 k =1

ai b j i

jk

ek .

(44.1)

Here ai and b j are the coordinates of the vectors a and b in the basis e1 , e2 , e3 . In order to simplify the formulas (44.1) note that the ma jority comp onents of the Levi-Civita symb ol are equal to zero. Only six of its twenty seven comp onents are nonzero. Applying the formula (43.5), we can bring the formula (44.1) to [a, b] = a1 b2 e3 + a2 b3 e1 + a3 b1 e2 - - a2 b1 e3 - a3 b2 e1 - a1 b3 e2 . [a, b] = e1 (a2 b3 - a3 b2 ) - Up on collecting similar terms the formula (44.2) yields (44.3) (44.2)

- e2 (a1 b3 - a3 b1 ) + e3 (a1 b2 - a2 b1 ). Now from the formula (44.3) we derive [a, b] = e a
1 2

a b

3

b

2

3

-e

a
2

1

a b

3

b

1

3

+e

a
3

1

a b

2

b

1

2

.

(44.4)

In deriving the formula (44.4) we used the formula for the determinant of a 2 â 2 matrix (see [7]). Note that the right hand side of the formula (44.4) is the expansion of the determinant of a 3 â 3 matrix by its first row (see [7]). Therefore this formula can b e written as: e b
1 1

e a b

2 2

e a b

3 3

[a, b] = a

.

(44.5)

1

2

3


104

CHAPTER I. VECTOR ALGEBRA.

Here a1 , a2 , a3 and b1 , b2 , b3 are the coordinates of the vectors a and b in the basis e1 , e2 , e3 . They fill the second and the third rows in the determinant (44.5) . Definition 44.1. The formulas (44.1) and (44.5) are called the formulas for calculating the vector product through the coordinates of vectors in a right orthonormal basis. Let's proceed to the case of a left orthonormal basis. In this case the structural constants of the vector product are given by the formula (43.7). Substituting (43.7) into (41.5) , we get
3 3 3

[a, b] = -

ai b j i
i =1 j =1 k =1

jk

ek .

(44.6)

Then from (44.6) we derive the formula e [a, b] = - a b
1 1

e a b

2 2

e a b

3 3

.

(44.7)

1

2

3

Definition 44.2. The formulas (44.6) and (44.7) are called the formulas for calculating the vector product through the coordinates of vectors in a left orthonormal basis. § 45. Mixed product. Definition 45.1. The mixed product of three free vectors a, b, and c is a numb er obtained as the scalar product of the vector a by the vector product of b and c: (a, b, c) = (a, [b, c]). (45.1)

As we see in the formula (45.1), the mixed product has three multiplicands. They are separated from each other by


§ 46. . . . IN AN ORTHONORMAL BASIS.

105

commas. Commas are the multiplication signs in writing the mixed product, not by themselves, but together with the round brackets surrounding the whole expression. Commas and brackets in writing the mixed product are natural delimiters for multiplicands: the first multiplicand is an expression b etween the op ening bracket and the first comma, the second multiplicand is an expression placed b etween two commas, and the third multiplicand is an expression b etween the second comma and the closing bracket. Therefore in complicated expressions no auxiliary delimiters are required. Foe example, in (a + b, c + d, e + f ) the sums a + b, c + d, and e + f are calculated first, then the mixed product itself is calculated. § 46. Calculation of the mixed product through the coordinates of vectors in an orthonormal basis. The formula (45.1) reduces the calculation of the mixed product to successive calculations of the vectorial and scalar products. In the case of the vectorial and scalar products we already have rather efficient formulas for calculating them through the coordinates of vectors in an orthonormal basis. Let e1 , e2 , e3 b e a right orthonormal basis and let a, b, and c b e free vectors given by their coordinates in this basis: a a a
1 2 3

a=

,

b=

b b b

1 2 3

,

c=

c1 c2 . c3

(46.1)

Let's denote d = [b, c]. Then the formula (45.1) is written as (a, b, c) = (a, d). (46.2)

In order to calculate the vector d = [b, c] we apply the formula
CopyRight c Sharipov R.A., 2010.


106

CHAPTER I. VECTOR ALGEBRA.

(44.1) which now is written as follows:
3 3 3

d=
k =1 i =1 j =1

b i c j i

jk

ek .

(46.3)

The formula (46.3) is an expansion of the vector d in the basis e1 , e2 , e3 . Hence the coefficients in this expansion should coincide with the coordinates of the vector d:
3 3

d=
i =1 j =1

k

b i c j ij k .

(46.4)

The next step consists in using the coordinates of the vector d from (46.4) for calculating the scalar product in the right hand side of the formula (46.2). The formula (33.3) now is written as
3

(a, d) =
k =1

ak d k .

(46.5)

Let's substitute (46.4) into (46.5) and take into account (46.2). As a result we get the formula
3 3 3

(a, b, c) =
i =1

a

k i =1 j =1

b i c j i

jk

.

(46.6)

Expanding the right hand side of the formula (46.6) and changing the order of summations in it, we bring it to
3 3 3

(a, b, c) =
i =1 j =1 k =1

b i c j i

jk

ak .

(46.7)

Note that the right hand side of the formula (46.7) differs from that of the formula (44.1) by changing ai for b i , changing b j for


§ 46. . . . IN AN ORTHONORMAL BASIS.

107

c j , and changing ek for ak . For this reason the formula (46.7) can b e brought to the following form analogous to (44.5): a (a, b, c) = b
1

a b

2

a b

3

1

2

3

.

(46.8)

c1

c2

c3

Another way for transforming the formula (46.7) is the use of the complete antisymmetry of the Levi-Civita symb ol (43.8). Applying this prop erty, we derive the identity ij k = k ij . Due to this identity, up on changing the order of multiplicands and redesignating the summation indices in the right hand side of the formula (46.7), we can bring this formula to the following form:
3 3 3

(a, b, c) =
i =1 j =1 k =1

ai b j c k ij k .

(46.9)

Definition 46.1. The formulas (46.8) and (46.9) are called the formulas for calculating the mixed product through the coordinates of vectors in a right orthonormal basis. The coordinates of the vectors a, b, and c used in the formulas (46.8) and (46.9) are taken from (46.1) . Let's proceed to the case of a left orthonormal basis. Analogs of the formulas (46.8) and (46.9) for this case are obtained by changing the sign in the formulas (46.8) and (46.9): a1 (a, b, c) = - b c
3 3 1 1

a2 b2 c
3 2

a3 b3 . c
3

(46.10)

(a, b, c) = -

ai b j c k ij k .
i =1 j =1 k =1

(46.11)


108

CHAPTER I. VECTOR ALGEBRA.

Definition 46.2. The formulas (46.10) and (46.11) are called the formulas for calculating the mixed product through the coordinates of vectors in a left orthonormal basis. The formulas (46.10) and (46.11) can b e derived using the theorem 42.2 or comparing the formulas (43.6) and (43.7). § 47. Prop erties of the mixed product. Theorem 47.1. The mixed product p ossesses the following four prop erties which are fulfilled for any four vectors a, b, c, d and for any numb er : 1) (a, b, c) = -(a, c, b), (a, b, c) = -(a, b, a), (a, b, c) = -(b, a, c); 2) (a + b, c, d) = (a, c, d) + (b, c, d); 3) ( a, c, d) = (a, c, d); 4) (a, b, c) = 0 if and only if the vectors a, b, and c are coplanar. Definition 47.1. The prop erty 1) expressed by three equalities in the theorem 47.1 is called the prop erty of complete skew symmetry or complete antisymmetry, the prop erties 2) and 3) are called the prop erties of linearity with respect to the first multiplicand, the prop erty 4) is called the vanishing condition. Proof of the theorem 47.1. The first of the three equalities comp osing the prop erty of complete antisymmetry 1) follows from the formula (45.1) and the theorems 39.1 and 28.1: (a, b, c) = (a, [b, c]) = (a, -[c, b]) = -(a, [c, b]) = -(a, c, b). The other two equalities entering the prop erty 1) cannot b e derived in this way. Therefore we need to use the formula (46.8). Transp osition of two vectors in the left hand side of this formula corresp onds to the transp osition of two rows in the determinant in the right hand side of this formula. It is well known that the


§ 47. PROPERTIES OF THE MIXED PRODUCT.

109

transp osition of any two rows in a determinant changes its sign. This observation proves all of the three equalities comp osing the prop erty of complete antisymmetry for the mixed product. The prop erties of linearity 2) and 3) of the mixed product in the theorem 47.1 are derived from the corresp onding prop erties of the scalar and vectorial products due to the formula (45.1) : (a + b, c, d) = (a + b, [c, d]) = (a, [c, d]) + + (b, [c, d]) = (a, c, d) + (b, c, d), ( a, c, d) = ( a, [c, d]) = (a, [c, d]) = (a, c, d). Let's proceed to proving the fourth prop erty of the mixed product in the theorem 47.1. Assume that the vectors a, b, and c are coplanar. In this case they are parallel to some plane in the space E and one can choose their geometric realizations lying on this plane . If b c, then the vector product d = [b, c] is nonzero and p erp endicular to the plane . As for the vector a, it is parallel to this plane. Hence d a, which yields the equalities (a, b, c) = (a, [b, c]) = (a, d) = 0. If b c, then the vector product [b, c] is equal to zero and the equality (a, b, c) = 0 is derived from [b, c] = 0 with use of the initial formula (45.1) for the scalar product. Now, conversely, assume that (a, b, c) = 0. If b c, then the vectors a, b, and c determine not more than two directions in the space E. For any two lines in this space always there is a plane to which these lines are parallel. In this case the vectors a, b, and c are coplanar regardless to the equality (a, b, c) = 0. If b c, then d = [b, c] = 0. Choosing geometric realizations of the vectors b and c with some common initial p oint O, we easily build a plane comprising b oth of these geometric realizations. The vector d = 0 in p erp endicular to this plane . Then from (a, b, c) = (a, [b, c]) = (a, d) = 0 we derive a d, which yields a . The vectors b and c are also parallel to the plane since their geometric realizations lie on this plane.


110

CHAPTER I. VECTOR ALGEBRA.

Hence all of the three vectors a, b, and c are parallel to the plane , which means that they are coplanar. The theorem 47.1 is completely proved. Theorem 47.2. Apart from the prop erties 1)­4), the mixed product p ossesses the following four prop erties which are fulfilled for any four vectors a, b, c, d and for any numb er : 5) 6) 7) 8) (c, (c, (c, (c, a+ a, d, a d, b, d) + a) d) = b) = = = (c, a, d) + (c, b, d); (c, a, d); (c, d, a) + (c, d, b); (c, d, a).

Definition 47.2. The prop erties 5) and 6) in the theorem 47.2 are called the prop erties of linearity with respect to the second multiplicand, the prop erties 7) and 8) are called the prop erties of linearity with respect to the third multiplicand. The prop erty 5) is derived from the prop erty 2) in the theorem 47.1 in the following way: (c, a + b, d) = -(a + b, c, d) = -((a, c, d) + (b, c, d)) = = -(a, c, d) - (b, c, d) = (c, a, d) + (c, b, d). The prop erty 1) from this theorem is also used in the ab ove calculations. As for the prop erties 6), 7), and 8) in the theorem 47.2, they are also easily derived from the prop erties 2) and 3) with the use of the prop erty 1). Indeed, we have (c, a, d) = -( a, c, d) = - (a, c, d) = (c, a, d), (c, d, a + b) = -(a + b, d, c) = -((a, d, c) + (b, d, c)) = = -(a, d, c) - (b, d, c) = (c, d, a) + (c, d, b), (c, d, a) = -( a, d, c) = - (a, d, c) = (c, d, a). The calculations p erformed prove the theorem 47.2.


§ 48. THE CONCEPT OF THE ORIENTED VOLUME.

111

§ 48. The concept of the oriented volume. Let a, b, c b e a right triple of non-coplanar vectors in the space E. Let's consider their mixed product (a, b, c). Due to the item 4) from the theorem 47.1 the non-coplanarity of the vectors a, b, c means (a, b, c) = 0, which in turn due to (45.1) implies [b, c] = 0. Due to the item 4) from the theorem 39.1 the non-vanishing condition [b, c] = 0 means b c. Let's build the geometric realizations of the non-collinear vectors b and c at some common initial p oint O and denote them - - - - b = OB and c = OC . Then we build the geometric realization of the vector a at the same initial - - p oint O and denote it through a = OA . Let's complement the - - - - - - vectors OA , OB , and OC up to a skew-angular parallelepip ed as shown in Fig. 48.1. Let's denote d = [b, c]. The vector d is p erp endicular to the base plane of the parallelepip ed, its length is calculated by the formula |d| = |b| |c| sin . It is easy to see that the length of d coincides with the base area of our parallelepip ed, i. e. with the area of the parallelogram built on the vectors b and c: S = |d| = |b| |c| sin . (48.1)

Theorem 48.1. For any two vectors the length of their vector product coincides with the area of the parallelogram built on these two vectors. The fact formulated in the theorem 48.1 is known as the geometric interpretation of the vector product.


112

CHAPTER I. VECTOR ALGEBRA.

Now let's return to Fig. 48.1. Applying the formula (45.1), for the mixed product (a, b, c) we derive (a, b, c) = (a, [b, c]) = (a, d) = |a| |d| cos . (48.2)

Note that |a| cos is the length of the segment [OF ], which coincides with the length of [AH ]. The segment [AH ] is parallel to the segment [OF ] and to the vector d, whivh is p erp endicular to the base plane of the skew-angular parallelepip ed shown in Fig. 48.1. Hence the segment [AH ] represents the height of this parallelepip ed ad we have the formula h = |AH | = |a| cos . Now from (48.1), (48.3), and (48.2) we derive (a, b, c) = S h = V , (48.4) (48.3)

i. e. the mixed product (a, b, c) in our case coincides with the volume of the skew-angular parallelepip ed built on the vectors a, b, and c. In the general case the value of the mixed product of three non-coplanar vectors a, b, c can b e either p ositive or negative, while the volume of a parallelepip ed is always p ositive. Therefore in the general case the formula (48.4) should b e written as V, if a, b, c is a right triple of vectors;

(a, b, c) =

Definition 48.1. The oriented volume of an ordered triple of non-coplanar vectors is a quantity which is equal to the volume of the parallelepip ed built on these vectors in the case where these vectors form a right triple and which is equal to the volume of
CopyRight c Sharipov R.A., 2010.

-V , if a, b, c is a left triple of vectors.

(48.5)


§ 49. STRUCTURAL CONSTANTS . . .

113

this parallelepip ed taken with the minus sign in the case where these vectors form a left triple. The formula (48.5) can b e written as a theorem. Theorem 48.2. The mixed product of any triple of non-coplanar vectors coincides with their oriented volume. Definition 48.2. If e1 , e2 , e3 is a basis in the space E, then the oriented volume of the triple of vectors e1 , e2 , e3 is called the oriented volume of this basis. § 49. Structural constants of the mixed product. Let e1 , e2 , e3 b e some basis in the space E. Let's consider various mixed products comp osed by the vectors of this basis: ci
jk

= (ei , ej , ek ).

(49.1)

Definition 49.1. For any basis e1 , e2 , e3 in the space E the quantities cij k given by the formula (49.1) are called the structural constants of the mixed product in this basis. The formula (49.1) is similar to the formula (29.6) for the comp onents of the Gram matrix. However, the structural constants of the mixed product cij k in (49.1) constitute a three index array which cannot b e laid into a matrix. An imp ortant prop erty of the structural constants cij k is their complete skew symmetry or complete antisymmetry. This prop erty is expressed by the following equalities: ci
jk

= -cj ik ,

ci

jk

= -cikj ,

ci

jk

= -ckj i ,

(49.2)

The relationships (49.2) mean that under the transp osition of any two indices the quantity cij k changes its sign. These relationships are easily derived from (43.5) by applying the item 1) from the theorem 47.1 to the right hand side of (49.1).


114

CHAPTER I. VECTOR ALGEBRA.

The following relationships are an immediate consequence of the prop erty of complete antisymmetry of the structural constants of the mixed product cij k : ci
ik

= -ci ik ,

ci

jj

= -cij j ,

ci

ji

= -cij i ,

(49.3)

They are derived by substituting j = i, k = j , and k = i into (49.2). From the relationships (49.3) the vanishing condition for the structural constants cij k is derived: ci
jk

= 0, if there are coinciding values of the indices i, j, k .

(49.4)

Now assume that the values of the indices i, j, k do not coincide. In this case, applying the relationships (49.2) , we derive ci ci
jk

= c123 if the indices i, j, k take the values (1, 2, 3), (2, 3, 1), or (3, 1, 2); = -c123 if the indices i, j, k take the values (1, 3, 2), (3, 2, 1), or (2, 1, 3).

(49.5) (49.6)

jk

The next step consists in comparing the relationships (49.4), (49.5), and (49.6) with the formula (43.5) that determines the Levi-Civita symb ol ij k . Such a comparison yields ci
jk

= c123 ij k .

(49.7)

Note that c123 = (e1 , e2 , e3 ). This formula follows from (49.1). Therefore the formula (49.7) can b e written as ci
jk

= (e1 , e2 , e3 ) ij k .

(49.8)

Theorem 49.1. In an arbitrary basis e1 , e2 , e3 the structural constants of the mixed product are expressed by the formula (49.8) through the only one constant -- the oriented volume of the basis.


§ 50. . . . IN A SKEW-ANGULAR BASIS.

115

§ 50. Calculation of the mixed product through the coordinates of vectors in a skew-angular basis. Let e1 , e2 , e3 b e a skew-angular basis in recall that the term skew-angular basis in th synonym of an arbitrary basis (see Definition c b e free vectors given by their coordinates in a= a a a
1 2 3

the space E. Let's is b ook is used as a 29.1). Let a, b, and the basis e1 , e2 , e3 : c1 c2 . c3 (50.1)

,

b=

b b b

1 2 3

,

c=

The formulas (50.1) mean that we have the expansions
3 3 3

a=
i =1

a ei ,

i

b=
j =1

b ej ,

j

c=
k =1

c k ek .

(50.2)

Let's substitute (50.2) into the mixed product (a, b, c):
3 3 3

(a, b, c) =
i =1

a ei ,
j =1

i

b ej ,
k =1

j

ck e

k

.

(50.3)

In order to tr of the mixed 47.2. Due to over i, j , and

ansform the formula (50.3) we apply the prop erties product 2), 5), and 7) from the theorems 47.1 and these prop erties we can bring the summation signs k outside the brackets of the mixed product:
3 3 3

(a, b, c) =
i =1 j =1 k =1

(ai ei , b j ej , c k ek ).

(50.4)

Now we apply the prop erties 3), 6), 8) from the theorems 47.1 and 47.2. Due to these prop erties we can bring the numeric factors ai , b j , c k outside the brackets of the mixed product (50.4):
3 3 3

(a, b, c) =
i =1 j =1 k =1

ai b j c k (ei , ej , ek ).

(50.5)


116

CHAPTER I. VECTOR ALGEBRA.

The quantities (ei , ej , ek ) are structural constants of the mixed product in the basis e1 , e2 , e3 (see (49.1)). Therefore the formula (50.5) can b e written as follows:
3 3 3

(a, b, c) =
i =1 j =1 k =1

ai b j c k cij k .

(50.6)

Let's substitute (49.8) into (50.6) oriented volume (e1 , e2 , e3 ) does indices i, j , and k. Therefore the b e brought outside the sums as a
3

and take into account that the not dep end on the summation oriented volume (e1 , e2 , e3 ) can common factor:
3 3

(a, b, c) = (e1 , e2 , e3 )
i =1 j =1 k =1

ai b j c k ij k .

(50.7)

Note that the formula (50.7) differs from the formula (46.9) only by the extra factor (e1 , e2 , e3 ) in its right hand side. As for the formula (46.9), it is brought to the form (46.8) by applying the prop erties of the Levi-Civita symb ol ij k only. For this reason the formula (50.7) can b e brought to the following form: a (a, b, c) = (e1 , e2 , e3 ) b
1

a b

2

a b

3

1

2

3

.

(50.8)

c1

c2

c3

Definition 50.1. The formulas (50.6), (50.7) , and (50.8) are called the formulas for calculating the mixed product through the coordinates of vectors in a skew-angular basis. § 51. The relation of structural constants of the vectorial and mixed products. The structural constants of the mixed product are determined by the formula (49.1). Let's apply the formula (45.1) in order to


§ 51. THE RELATION OF STRUCTURAL CONSTANTS . . .

117

transform (49.1). As a result we get the formula ci
jk

= (ei , [ej , ek ]).

(51.1)

Now we can apply the formula (40.2). Let's write it as follows:
3

[ej , ek ] =
q =1

C

q jk

eq .

(51.2)

Substituting (51.2) into (51.1) and taking into account the properties 5) and 6) from the theorem 28.2, we derive
3

ci

jk

=
q =1

C

q jk

(ei , eq ).

(51.3)

Let's apply the formulas (29.6) and (30.1) to (51.3) and bring the formula (51.3) to the form
3

ci

jk

=
q =1

C

q jk

gq i .

(51.4)

The following formula is somewhat more b eautiful:
3

ci

jk

=
q =1

q Cij gqk .

(51.5)

In order to derive the formula (51.5) we apply the identity cij k = cj k i to the left hand side of the formula (51.4). This identity is derived from (49.2). Then we p erform the cyclic redesignation of indices i k j i. The formula (51.5) is the first formula relating the structural constants of the vectorial and mixed products. It is imp ortant from the theoretical p oint of view, but this formula is of little


118

CHAPTER I. VECTOR ALGEBRA.

use practically. Indeed, it expresses the structural constants of the mixed product through the structural constants of the vector product. But for the structural constants of the mixed product we already have the formula (49.8) which is rather efficient. As for the structural constants of the vector product, we have no formula yet, except the initial definition (40.2). For this reason q we need to invert the formula (51.5) and express Cij through cij k . In order to reach this goal we need some auxiliary information on the Gram matrix. Theorem 51.1. The Gram matrix G of any basis in the space E is non-degenerate, i. e. its determinant is nonzero: det G = 0. Theorem 51.2. For any basis e1 , e2 , e3 in the space E the determinant of the Gram matrix G is equal to the square of the oriented volume of this basis: det G = (e1 , e2 , e3 )2 . Th e sis is a of the get (e1 (51.6)

theorem 51.1 follows from the theorem 51.2. Indeed, a batriple of non-coplanar vectors. From the non-coplanarity vectors e1 , e2 , e3 due to item 4) of the theorem 47.1 we , e2 , e3 ) = 0. Then the formula (51.6) yields det G > 0, (51.7)

while the theorem 51.1 follows from the inequality (51.7). I will not prove the theorem 51.2 right now at this place. This theorem is proved b elow in § 56. Let's proceed to deriving consequences from the theorem 51.1. It is known that each non-degenerate matrix has an inverse matrix (see [7]). Let's denote through G-1 the matrix inverse to the Gram matrix G. In writing the comp onents of the matrix G-1 the following convention is used. Let's proceed to deriving consequences from the theorem 51.1. It is known that each non-degenerate matrix has an inverse matrix (see [7]). Let's


§ 51. THE RELATION OF STRUCTURAL CONSTANTS . . .

119

denote through G-1 the matrix inverse to the Gram matrix G. In writing the comp onents of the matrix G-1 the following convention is used. Let's proceed to deriving consequences from the theorem 51.1. It is known that each non-degenerate matrix has an inverse matrix (see [7]). Let's denote through G-1 the matrix inverse to the Gram matrix G. In writing the comp onents of the matrix G-1 the following convention is used. Definition 51.1. For denoting the comp onents of the matrix G-1 inverse to the Gram matrix G the same symb ol g as for the comp onents of the matrix G itself is used, but the comp onents of the inverse Gram matrix are enumerated with two upp er indices: g G-1 = g g
11 21 31

g g g

12 22 32

g g g

13 23 33

(51.8)

The matrices G and G-1 are inverse to each other. Their product in any order is equal to the unit matrix: G · G-1 = 1, G-1 · G = 1. (51.9)

From the regular course of algebra we know that each of the equalities (51.9) fixes the matrix G-1 uniquely once the matrix G is given (see. [7]). Now we apply the matrix transp osition op eration to b oth sides of the matrix equalities (51.9): (G · G-1 ) = 1 = 1. (51.10)

Then we use the identity (A · B ) = B · A from the exercise 29.2 in order to transform the formula (51.10) and take into account the symmetry of the matrix G (see Theorem 30.1): (G-1 ) · G = (G-1 ) · G = 1.
CopyRight c Sharipov R.A., 2010.

(51.11)


120

CHAPTER I. VECTOR ALGEBRA.

The rest is to compare the equality (51.11) with the second matrix equality (51.9) . This yields (G-1 ) = G-
1

(51.12)

The formula (51.12) can b e written as a theorem. Theorem 51.3. For any basis e1 , e2 , e3 in the space E the matrix G-1 inverse to the Gram matrix of this basis is symmetric. In terms of the matrix comp onents of the matrix (51.7) the equality (51.12) is written as an equality similar to (30.1): g
ij

= g j i.

(51.13)

The second equality (51.9) is written as the following relationships for the comp onents of the matrices (51.8) and (29.7):
3

g
k =1

sk

s gkq = q .

(51.14)

s Here q is the Kronecker symb ol defined by the formula (23.3). Using the symmetry identities (51.13) and (30.1) , we write the relationship (51.14) in the following form: 3

gqk g
k =1

ks

s = q .

(51.15)

Now let's multiply b oth ides of the equality (51.5) by gks and then p erform summation over k in b oth sides of this equality:
3 3 3 q Cij gqk gks . k =1 q =1

ci
k =1

jk

g

ks

=

(51.16)

If we take into account the identity (51.15), then we can bring


§ 52. EFFECTIVIZATION OF THE FORMULAS . . .

121

the formula (51.16) to the following form:
3 3

ci
k =1

jk

g

ks

=
q =1

qs Cij q .

(51.17)

When summing over q in the right hand side of the equality (51.17) the index q runs over three values 1, 2, 3. Then the s Kronecker symb ol q takes the values 0 and 1, the value 1 is taken only once when q = s. This means that only one of three summands in the right hand side of (51.17) is nonzero. This s nonzero summand is equal to Cij . Hence the formula (51.17) can b e written in the following form:
3

ci
k =1

jk

g

ks

s = Ci j .

(51.18)

Let's change the symb ol k for q and the symb ol s for k in (51.18). Then we transp ose left and right hand sides of this formula:
3 k Ci j = q =1

cij q gqk .

(51.19)

The formula (51.19) is the second formula relating the structural constants of the vectorial and mixed products. On the base of the relationships (51.5) and (51.19) we formulate a theorem. Theorem 51.4. For any basis e1 , e2 , e3 in the space E the structural constants of the vectorial and mixed products in this basis are related to each other in a one-to-one manner by means of the formulas (51.5) and (51.19) . § 52. Effectivization of the formulas for calculating vectorial and mixed products. Let's consider the formula (29.8) for calculating the scalar product in a skew-angular basis. Apart from the coordinates of


122

CHAPTER I. VECTOR ALGEBRA.

vectors, this formula uses the comp onents of the Gram matrix (29.7). In order to get the comp onents of this matrix one should calculate the mutual scalar products of the basis vectors (see formula (29.6)), for this purp ose one should measure their lengths and the angles b etween them (see Definition 26.1). No other geometric constructions are required. For this reason the formula (29.8) is recognized to b e effective. Now let's consider the formula (41.5) for calculating the vector product in a skew-angular basis. This formula uses the structural constants of the vector product which are defined by means of the formula (40.2). According to this formula, in order to calculate the structural constants one should calculate the vector products [ei , ej ] in its left hand side. For this purp ose one should construct the normal vectors (p erp endiculars) to the planes given by various pairs of basis vectors ei , ej . (see Definition 35.1). Up on calculating the vector products [ei , ej ] one should expand them in the basis e1 , e2 , e3 , which require some auxiliary geometric constructions (see formula (18.4) and Fig. 18.1). For this reason the efficiency of the formula (41.5) is much less than the efficiency of the formula (29.8). And finally we consider the formula (50.7) for calculating the mixed product in a skew-angular basis. In order to apply this formula one should know the value of the mixed product of the three basis vectors (e1 , e2 , e3 ). It is called the oriented volume of a basis (see Definition 48.2). Due to the theorem 48.2 and the definition 48.1 for this purp ose one should calculate the volume of the skew-angular parallelepip ed built on the basis vectors e1 , e2 , e3 . In order to calculate the volume of this parallelepip ed one should know its base area and its height. The area of its base is effectively calculated by the lengths of two basis vectors and the angle b etween them (see formula (48.1)). As for the height of the parallelepip ed, in order to find it one should drop a p erp endicular from one of its vertices to its base plane. Since we need such an auxiliary geometric construction, the formula (50.7) is less effective as compared to the formula (29.8) in the case of


§ 52. EFFECTIVIZATION OF THE FORMULAS . . .

123

the scalar product. In order to make the formulas (41.5) and (50.7) effective we use the formula (51.6) . It leads to the following relationship: (e1 , e2 , e3 ) = ± det G. (52.1) The sign in (52.1) is determin det G (e1 , e2 , e3 ) = - det G (a, b, c) = ± det G ed by the orientation of a basis: if the basis e1 , e2 , e is right;
3

(52.2) if the basis e1 , e2 , e3 is left. Let's substitute the expression (52.1) into the formula for the mixed product (50.7) . As a result we get
3 3 3

ai b j c k ij k .
i =1 j =1 k =1

(52.3)

Similarly, substituting (52.1) into the formula (50.8), we get a1 (a, b, c) = ± det G b1 c1 a b
2

a b

3

2

3

.

(52.4)

c2

c3

Definition 52.1. The formulas (52.3) and (52.4) are called the effectivized formulas for calculating the mixed product through the coordinates of vectors in a skew-angular basis. In the case of the formula (41.5), in order to make it effective we need the formulas (49.8) and (51.19) . Substituting the expression (52.1) into these formulas, we obtain cij k = ± det G ij k , (52.5) C = ± det G
k ij 3

ij q g qk .
q =1

(52.6)


124

CHAPTER I. VECTOR ALGEBRA.

Definition 52.2. The formulas (52.5) and (52.6) are called the effectivized formulas for calculating the structural constants of the mixed and vectorial products. Now let's substitute the formula (52.6) into (41.5). This leads to the following formula for the vector product: [a, b] = ± det G
3 3 3 3

ai b j ij q g
i =1 j =1 k =1 q =1

qk

ek .

(52.7)

Definition 52.3. The formula (52.7) is called the effectivized formula for calculating the vector product through the coordinates of vectors in a skew-angular basis. § 53. Orientation of the space. Let's consider the effectivized formulas (52.1), (52.3), (52.4), (52.5), (52.6), and (52.7) from § 52. Almost all information on a basis in these formulas is given by the Gram matrix G. The Gram matrix arising along with the choice of a basis reflects an imp ortant prop erty of the space E -- its metric. Definition 53.1. The metric of the space E is its structure (its feature) that consists in p ossibility to measure the lengths of segments and the numeric values of angles in it. The only non-efficiency remaining in the effectivized formulas (52.1), (52.3) , (52.4), (52.5), (52.6), and (52.7) is the choice of sign in them. As was said ab ove, the sign in these formulas is determined by the orientation of a basis (see formula (52.2)). There is absolutely no p ossibility to determine the orientation of a basis through the information comprised in its Gram matrix. The matter is that the mathematical space E describ ed by Euclid's axioms (see [6]), comprises the p ossibility to distinguish a pair of bases with different orientations from a pair of bases with coinciding orientations. However, it does not contain any reasons for to prefer bases with one of two p ossible orientations.


§ 54. CONTRACTION FORMULAS.

125

The concept of right triple of vectors (see Definition 34.2) and the p ossibility to distinguish right triples of vectors from left ones is due to the presence of p eople, it is due to their ability to observe vectors and compare their rotation with the rotation of clock hands. Is there a fundamental asymmetry between left and right not dep ending on the presence of p eople and on other nonfundamental circumstances? Is the space fundamental ly oriented? This is a question on the nature of the physical space E. Some research in the field of elementary particle physics says that such an asymmetry does exist. As for me, as the author of this b ook I cannot definitely assert that this problem is finally resolved. § 54. Contraction formulas. Contraction formulas is a collection of four purely algebraic identities relating the Levi-Civita symb ol and the Kronecker symb ol with each other. Theorem 54.1. The Levi-Civita symb ol and the Kronecker symb ol are related by the first contraction formula
m i m j n j p j m k n k . p k

m

np

i

jk

n = i p i

(54.1)

n Proof. Let's denote through fimk p the right hand side of j the first contraction formula (54.1) . The transp osition of any n two lower indices in fimk p is equivalent to the corresp onding j transp osition of two columns in the matrix (54.1). It is known that the transp osition of two columns of a matrix results in changing the sign of its determinant. This yield the following n relationships for the quantities fimk p : j

f

m np ij k

= -f

m np j ik

,

f

m np ij k

= -f

m np ik j

,

f

m np ij k

= -f

m np kj i

.

(54.2)


126

CHAPTER I. VECTOR ALGEBRA.

The relationships (54.2) are analogous to the relationships (49.2). Rep eating the considerations used in § 49 when deriving the formulas (49.4), (49.5), (49.6), from the formula (54.2) we derive 0 f
m np 123

if there are coinciding values of the indices i, j, k ; if the values of the indices i, j, k form an even p ermutation of the numb ers 1, 2, 3; , if the values of the indices i, j, k form an odd p ermutation of the numb ers 1, 2, 3. (54.3)

f

m np ij k

=

Let's compare the formula (54.3) with the formula (43.5) defining the Levi-Civita symb ol. Such a comparison yields f
m np ij k

-f

m np 123

=f

m np 123 ij k

.
m np 123

(54.4) in (54.4) is

n Like the initial quantities fimk p , the factor f j defined as the determinant of a matrix: m 1 m 2 n 2 p 2 m 3 n 3 . p 3

f

m np 123

n = 1 p 1

(54.5)

mn Transp osition of any pair of the upp er indices in f123 p is equivalent to the corresp onding transp osition of rows in the matrix (54.5). Again we know that the transp osition of any two rows of a matrix changes the sign of its determinant. Hence we derive the following relationships for the quantities (54.5):

f

m np 123

= -f

nm p 123

,

f

m np 123

= -f

mp n 123

,

f

m np 123

= -f

p nm 123

(54.6)

The relationships (54.6) are analogous to the relationships (54.2), which are analogous to (49.2). Rep eating the considerations used
CopyRight c Sharipov R.A., 2010.


§ 54. CONTRACTION FORMULAS.

127

in § 49, from the relationships (54.6) we immediately derive the formula analogous to the formula (54.3): 0 f 123 123 -f if there are coinciding values of the indices m, n, p; if the values of the indices m, n, p form an even p ermutation of the numb ers 1, 2, 3; if the values of the indices m, n, p form an odd p ermutation of the numb ers 1, 2, 3. (54.7)

f

m np 123

=

123 123

Let's compare the formula (54.7) with the formula (43.5) defining the Levi-Civita symb ol. This comparison yields f
m np 123

=f

123 mnp 123

.

(54.8)

Let's combine the formulas (54.4) and (54.8), i. e. we substitute (54.8) into (54.4). This substitution leads to the formula f
m np ij k

=f

123 mnp ij k 123

.

(54.9)

123 Now we need to calculate the coefficient f123 in the formula m np (54.9). Let's recall that the quantity fij k was defined as the right hand side of the formula (54.1) . Therefore we can write m i m j n j p j m k n k p k

f

m np ij k

n = i p i

(54.10)

and substitute i = m = 1, j = n = 2, k = p = 3 into (54.10) :
1 1 1 2 2 2 3 2 1 3

f

123 123

2 = 1 3 1

100 2 3 = 0 1 0 = 1. 3 001 3

(54.11)


128

CHAPTER I. VECTOR ALGEBRA.

123 Taking into account the value of the coefficient f123 given by the formula (54.11), we can transform the formula (54.9) to

f

m np ij k

= m

np

ij k .

(54.12)

Now the required contraction formula (54.1) is obtained as a consequence of (54.10) and (54.12). The theorem 54.1 is proved. Theorem 54.2. The Levi-Civita symb ol and the Kronecker symb ol are related by the second contraction formula
3


k =1

m nk

i

jk

=

m i n i

m j n j

.

(54.13)

Proof. The second contraction formula (54.13) is derived from the first contraction formula (54.1). For this purp ose we substitute p = k into the formula (54.1) and take into account k that k = 1. This yields the formula
m i m j n j k j m k n k .

m

nk

i

jk

n = i k i

(54.14)

1

Let's insert the summation over k to b oth sides of the formula (54.14). As a result we get the formula
3 3 m i n i k =1 k i m j n j k j m k n k .

m
k =1

nk

i

jk

=

(54.15)

1

The left hand side of the with the left hand side of the to b e proved. For this reason side of the formula (54.15) on

obtained formula (54.15) coincides second contraction formula (54.13) b elow we transform the right hand ly. Let's expand the determinant in


§ 54. CONTRACTION FORMULAS.

129

the formula (54.15) by its last row:
3 3


k =1

m nk

i

jk

=
k =1 m i n i

k i m k n k

m j n j

m k n k m i n i

-
m j n j

(54.16) .

k - j

+

The summation over k in the right hand side of the formula (54.16) applies to all of the three terms enclosed in the round brackets. Expanding these brackets, we get
3 3

m
k =1

nk

i

jk

=
k =1 m i n i

k i m k n k

m j n j 3

m k n k

-
m j n j

3

-

k j k =1

+
k =1

m i n i

(54.17) .

The first sum in the right hand side of (54.17) contains the k factor i . In p erforming the summation cycle over k this factor app ears to b e nonzero only once when k = i. For this reason only one summand of the first sum does actually survive. In this term k = i. Similarly, in the second sum also only one its term does actually survive, in this term k = j . As for the last sum in the right hand side of (54.17), the expression b eing summed does not dep end on k. Therefore it triples up on calculating this sum. j i Taking into account that i = j = 1, we get
3

m
k =1

nk

i

jk

=

m j n j

m i n i

-

m i n i

m j n j

+3

m i n i

m j n j

.

The first determinant in the right hand side of the ab ove formula differs from two others by the transp osition of its columns. If we p erform this transp osition once more, it changes its sign and


130

CHAPTER I. VECTOR ALGEBRA.

we get a formula with three coinciding determinants. collecting the similar terms, we derive
3

Then,

m
k =1

nk

i

jk

= (-1 - 1 + 3)

m i n i

m j n j

.

(54.18)

Now it is easy to see that the formula (54.18) leads to the required formula (54.13). The theorem 54.2 is proved. Theorem 54.3. The Levi-Civita symb ol and the Kronecker symb ol are related by the third contraction formula
3 3

m
j =1 k =1

jk

i

jk

m = 2 i .

(54.19)

Proof. The third contraction formula the second contraction formula (54.13) . substitute n = j into the formula (54.13), j j = 1, and insert the summation over j obtained equality. This yields the formula
3 3 3

(54.19) is derived from For this purp ose let's take into account that into b oth sides of the

m
j =1 k =1

jk

i

jk

=
j =1

m i j i

m j

3

1

=
j =1

m mj (i - j i ).

Up on expanding the brackets the sum over j in the right hand side of the ab ove formula can b e calculated explicitly:
3 3 3 3 m i j =1


j =1 k =1

mj k

i

jk

=

-

j =1

mj m m m j i = 3 i - i = 2 i .

It is easy to see that the calculations p erformed prove the formula (54.19) and the theorem 54.3 in whole.


§ 55. . . . AND THE JACOBI IDENTITY.

131

Theorem 54.4. The Levi-Civita symb ol and the Kronecker symb ol are related by the fourth contraction formula
3 3 3

i
i =1 j =1 k =1

jk

i

jk

= 6.

(54.20)

Proof. The fourth contraction formula (54.20) is derived from the third contraction formula (54.19) . For this purp ose we substitute m = i into (54.19) and insert the summation over i into b oth sides of the obtained equality. This yields
3 3 3 3 3 i i = 2 i =1 i =1

i
i =1 j =1 k =1

jk

i

jk

=2

1 = 2 · 3 = 6.

The ab ove calculations prove the formula (54.20) , which completes the proof of the theorem 54.4. § 55. The triple product expansion formula and the Jacobi identity. Theorem 55.1. For any triple of free vectors a, b, and c in the space E the following identity is fulfilled: [a, [b, c]] = b (a, c) - c (a, b), which is known as the triple product expansion formula1 . Proof. In order to prove the identity (55.1) we choose some right orthonormal basis e1 , e2 , e3 in the space E and let's expand the vectors a, b, and c in this basis:
3 3 3

(55.1)

a=
i =1

ai ei ,

b=
j =1

b j ej ,

c=
k =1

c k ek .

(55.2)

1 In Russian literature the triple pro duct expansion formula is known as the double vectorial pro duct formula or the «BAC minus CAB» formula.


132

CHAPTER I. VECTOR ALGEBRA.

Let's denote d = [b, c] and use the formula (44.1) for to calculate the vector d. We write this formula as
3 3 3

d=
k =1 i =1 j =1

b i c j i

jk

ek .

(55.3)

The formula (55.3) is the expansion of the vector d in the basis e1 , e2 , e3 . Therefore we can get its coordinates:
3 3

dk =
i =1 j =1

b i c j ij k .

(55.4)

Now we again apply the formula (44.1) in order to calculate the vector [a, [b, c]] = [a, d]. In this case we write it as follows:
3 3 3

[a, d] =
n= 1 m = 1 k = 1

am dk m

kn

en .

(55.5)

Substituting (55.4) into the formula (55.5) , we get
3 3 3 3 3

[a, d] =
n= 1 m = 1 k = 1 i = 1 j = 1 3 3 3 3

am b i c j i
3

j k mk n

en = (55.6)

=
n= 1 m = 1 i = 1 j = 1

am b i c

j k =1

i

j k mk n

en .

The upp er or lower p osition of indices in the Levi-Civita symb ol does not matter (see (43.5)). Therefore, taking into account (43.8), we can write mkn = mkn = -mnk and bring the formula (55.6) to the following form:
3 3 3 3 3

[a, d] = -

a bc
n= 1 m = 1 i = 1 j = 1

m

i

j k =1

m

nk

i

jk

en .

(55.7)


§ 55. . . . AND THE JACOBI IDENTITY.

133

The sum enclosed in the round brackets in (55.7) coincides with the left hand side of the second contraction formula (54.13). Applying (54.13), we continue transforming the formula (55.7):
3 3 3 3

[a, d] = -
3 3

am b i c
n= 1 m = 1 i = 1 j = 1 3 3

j

m i n i

m j n j

en =

=
n= 1 m = 1 i = 1 j = 1 3 3 3 3

nm mn am b i c j i j - i j en =

=
n= 1 m = 1 i = 1 j = 1 3

nm am b i c j i j en - 3 3 3 mn am b i c j i j en =

-
3 3

n= 1 m = 1 i = 1 j = 1 3 3

=
i =1 j =1

aj b i c j ei -

ai b i c j e j .
i =1 j =1

The result obtained can b e written as follows:
3 3 3 3

[a, [b, c]] =
j =1

aj c

j i =1

bi e

i

-

ai b
i =1

i j =1

cj e

j

.

Mow basis is is easily account

let's recall that the scalar product in an orthonormal calculated by the formula (33.3). The formula (33.3) recognized within the ab ove relationship. Taking into this formula, we get the equality
3 3

[a, [b, c]] = (a, c)
i =1

b i ei - (a, b)

c j ej .
j =1

(55.8)

In order to bring (55.8) to the ultimate form (55.1) it is sufficient to find the expansions of the form (55.2) for b and c within the
CopyRight c Sharipov R.A., 2010.


134

CHAPTER I. VECTOR ALGEBRA.

ab ove formula (55.8). As a result the formula (55.8) takes the form [a, [b, c]] = (a, c) b - (a, b) c, which in essential coincides with (55.1). The theorem 55.1 is proved. Theorem 55.2. For any triple of free vectors a, b, and c in the space E the Jacobi identity is fulfilled: [a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0. (55.9)

Proof. The Jacoby identity (55.9) is easily derived with the use of the triple product expansion formula (55.1) : [a, [b, c]] = b (a, c) - c (a, b). (55.10)

Let's twice p erform the cyclic redesignation of vectors a b c a in the ab ove formula (55.10) . This yields [c, [a, b]] = a (c, b) - b (c, a). By adding the ab ove three equalities in the left hand side we get the req [b, [c, a]] + [c, [a, b]], while th right equality vanishes. The theorem 55.2 [b, [c, a]] = c (b, a) - a (b, c), (55.11) (55.12)

(55.10), (55.11), and (55.12) uired expression [a, [b, c]] + hand side of the resulting is proved.

§ 56. The product of two mixed products. Theorem 56.1. For any six free vectors a, b, c, x, y, and z in the space E the following formula for the product of two mixed products (a, b, c) and (x, y, z) is fulfilled: (a, x) (a, b, c) (x, y, z) = (a, y) (a, z) (b, x) (b, y) (b, z) (c, x) (c, y) . (c, z) (56.1)


§ 56. . . . OF TWO MIXED PRODUCTS.

135

In order to prove the formula (56.1) we need two prop erties of the matrix determinants which are known as the linearity with respect to a row and the linearity with respect to a column (see [7]). The first of them can b e expressed by the formula x . . .
k 1 1 1

r

... . . . ... . . . ...

x . . .
k n

1 n r

1 x1 . . .

... . . .

r

x1 n . . . i xk (i) . n

i =1

i x (i) . . . x
n 1

x (i) = . . . x
n n

i xk (i) . . . 1
i =1 i =1

. . . xn 1

. . . ...

. . . n xn

Lemma 56.1. If the k-th row of a square matrix is a linear combination of some r rows (non necessarily coinciding with the other rows of this matrix), then the determinant of such a matrix is equal to a linear combination of r separate determinants. The linearity with resp ect to a column is formulated similarly. Lemma 56.2. If the k-th column of a square matrix is a linear combination of some r columns (non necessarily coinciding with the other columns of this matrix), then the determinant of such a matrix is equal to a linear combination of r separate determinants. I do not write the formula illustrating the lemma 56.2 since it does not fit the width of a page in this b ook. This formula can b e obtained from the ab ove formula by transp osing the matrices in b oth its sides. Proof of the theorem 56.1. Let e1 , e2 , e3 b e some right orthonormal basis in the space E. Assume that the vectors a, b, c, x, y, and z are given by their coordinates in this basis. Let's denote through L the left hand side of the formula (56.1): L = (a, b, c) (x, y, z). (56.2)


136

CHAPTER I. VECTOR ALGEBRA.

In order to calculate L we apply the formula (46.9) . In the case of the vectors a, b, c the formula (46.9) yields
3 3 3

(a, b, c) =
i =1 j =1 k =1

ai b j c k ij k .

(56.3)

In the case of the vectors x, y, z the formula (46.9) is written as
3 3 3

(x, y, z) =
m=1 n=1 p=1

xm y n z p mnp .

(56.4)

Note that raising indices of the Levi-Civita symb ol in (56.4) does not change its values (see (43.5)). Now, multiplying the formulas (56.3) and (56.4), we obtain the formula for L:
3 3 3 3 3 3

L=
i=1 j =1 k =1 m=1 n=1 p=1

ai b j c k xm y n z p m

np

ij k .

(56.5)

The product mnp ij k in (56.5) can b e replaced by the matrix determinant taken from the first contraction formula (54.1):
3 3 ... m i m j n j p j m k n k . p k

L=

ai b j c k xm y n z

p

n i p i

(56.6)

i =1 j =1 k =1 m=1 n=1 p=1

The next step consists in applying the lemmas 56.1 and 56.2 in order to transform the formula (56.6). Applying the lemma 56.2, we bring the sum over i and the associated factor ai into the first column of the determinant. Similarly, we bring the sum over j and the factor b j into the second column, and finally, we bring the sum over k and its associated factor c k into the third column of the determinant. Then we apply the lemma 56.1 in order to distribute the sums over m, n, and p to the rows of the


§ 56. . . . OF TWO MIXED PRODUCTS.

137

determinant. Simultaneously, we distribute the associated factors xm , y n , and z p to the rows of the determinant. As a result of our efforts we get the following formula:
3 3 m ai xm i i =1 m =1 3 3 i n n i j = 1 n= 1 3 p ai z p i i=1 p=1 j =1 p=1 3 p b j z p j k =1 p=1 j =1 m =1 3 3 j n n j k = 1 n= 1 3 3 p c k z p k 3 3 m b j xm j k =1 m =1 3 3 n c k y n k 3 3 m c k xm k

L=
i = 1 n= 1 3 3

ay

by

.

Due to the Kronecker symb ols the double sums in the ab ove formula are reduced to single sums:
3 3 3

ai x
i =1 3 i

i j =1 3

bj x

j k =1 3

ck x

k

L=
i =1 3

ay

i j =1 3

by

j

j k =1 3

ck y

k

.

(56.7)

ai z
i =1

i j =1

bj z

j k =1

ck z

k

Let's recall that our basis e1 , e2 , e3 is orthonormal. The scalar product in such a basis is calculated according to the formula (33.3). Comparing (33.3) with (56.7) , we see that all of the sums within the determinant (56.7) are scalar products of vectors: (a, x) L = (a, y) (a, z) (b, x) (b, y) (b, z) (c, x) (c, y) . (c, z) (56.8)


138

CHAPTER I. VECTOR ALGEBRA.

Now the formula (56.1) follows from the formulas (56.2) and (56.8). The theorem 56.1 is proved. The formula (56.1) is valuable for us not by itself, but due to its consequences. Assume that e1 , e2 , e3 is some arbitrary basis. Let's substitute a = e1 , b = e2 , c = e3 , x = e1 , y = e2 , z = e3 into the formula (56.1). As a result we obtain (e1 , e1 ) (e1 , e2 , e3 )2 = (e1 , e2 ) (e1 , e3 ) (e2 , e1 ) (e3 , e1 ) (e2 , e2 ) (e3 , e2 ) . (e2 , e3 ) (e3 , e3 ) (56.9)

Comparing (56.9) with (29.7) and (29.6), we see that the matrix (56.9) differs from the Gram matrix G by transp osing. If we take into account the symmetry of the Gram matrix (see theorem 30.1), than we find that they do coincide. Therefore the formula (56.9) is written as follows: (e1 , e2 , e3 )2 = det G. (56.10)

The formula (56.10) coincides with the formula (51.6). This fact proves the theorem 51.2, which was unproved in § 51.


CHAP TE R I I

GEOMETRY OF LINES AND SURFACES.

In this Chapter I E and for by these p

Chapter the tools of the vector algebra develop ed in are applied for describing separate p oints of the space describing some geometric forms which are comp osed oints. § 1. Cartesian coordinate systems.

Definition 1.1. A Cartesian coordinate system in the space E is a basis e1 , e2 , e3 complemented by some fixed p oint O of this space. The p oint O b eing a part of the Cartesian coordinate system O, e1 , e2 , e3 is called the origin of this coordinate system. Definition 1.2. The vector - - OX binding the origin O of a Cartesian coordinate system O, e1 , e2 , e3 with a p oint X of the space E is called the radius vector of the p oint X in this coordinate system. The free vectors e1 , e2 , e3 b eing constituent parts of a Cartesian coordinate system O, e1 , e2 , e3 remain free. However they are often represented by geometric realizations attached to the origin O (see Fig. 1.1). These geometric realizations are extended


140

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

up to the whole lines which are called the coordinate axes of the Cartesian coordinate system O, e1 , e2 , e3 . Definition 1.3. The coordinates of a p oint X in a Cartesian coordinate system O, e1 , e2 , e3 are the coordinates of its radius - - vector rX = OX in the basis e1 , e2 , e3 . Like other vectors, radius vectors of p oints are covered by the index setting convention (see Definition 20.1 in Chapter I). The coordinates of radius vectors are enumerated by upp er indices, they are usually arranged into columns: x x x
1 2 3

rX =

.

(1.1)

However, in those cases its coordinates these coor separated row and placed symb ol denoting the p oint

where a p oint X is represented by dinates are arranged into a commawithin round brackets just after the X itself: (1.2)

X = X (x1 , x2 , x3 ).

The upp er p osition of indices in the formula (1.2) is inherited from the formula (1.1). Cartesian coordinate systems can b e either rectangular or skew-angular dep ending on the typ e of basis used for defining them. In this b ook I follow a convention similar to that of the definition 29.1 in Chapter I. Definition 1.4. In this b ook a skew-angular coordinate system is understood as an arbitrary coordinate system where no restrictions for the basis are imp osed. Definition 1.5. A rectangular coordinate system O, e1 , e2 , e3 whose basis is orthonormal is called a rectangular coordinate system with unit scales along the axes.
CopyRight c Sharipov R.A., 2010.


§ 2. EQUATIONS OF LINES AND SURFACES.

141

- - A remark. The radius vector of a p oint X written as OX is a geometric vector whose p osition in the space is fixed. The radius vector of a p oint X written as rX is a free vector We can p erform various op erations of vector algebra with this vector: we can add it with other free vectors, multiply it by numb ers, comp ose scalar products etc. But the vector rX has a sp ecial mission -- to b e a p ointer to the p oint X . It can do this mission only if its initial p oint is placed to the origin. Exercise 1.1. Relying on the definition 1.1, formulate analogous definitions for Cartesian coordinate systems on a line and on a plane. § 2. Equations of lines and surfaces. Coordinates of a single fixed p oint X in the space E are three fixed numb ers (three constants). If these coordinates are changing, we get a moving p oint that runs over some set within the space E. In this b ook we consider the cases where this set is some line or some surface. The case of a line differs from the case of a surface by its dimension or, in other words, by the number of degrees of freedom. Each surface is two-dimensional -- a p oint on a surface has two degrees of freedom. Each line is one-dimensional -- a p oint on a line has one degree of freedom. Lines and surfaces contain infinitely many p oints. Therefore they cannot b e describ ed by enumeration where each p oint is describ ed separately. Lines and surfaces are describ ed by means of equations. The equations of lines and surfaces are sub divided into several typ es. If the radius vector of a p oint enters an equation as a whole without sub dividing it into separate coordinates, such an equation is called a vectorial equation. If the radius vector of a p oint enters an equation through its coordinates, such an equation is called a coordinate equation. Another attribute of the equations is the method of implementing the degrees of freedom. One or two degrees of freedom can b e implemented in an equation explicitly when the radius


142

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

vector of a p oint is given as a function of one or two variables, which are called parameters. In this case the equation is called parametric. Non-parametric equations b ehave as obstructions decreasing the numb er of degrees of freedom from the initial three to one or two. § 3. A straight line on a plane. Assume that some plane in the space E is chosen and fixed. Then the numb er of degrees of freedom of a p oint immediately decreases from three to two. In order to study various forms of equations defining a straight line on the plane we choose some coordinate system O, e1 , e2 on this plane. Then we can define the p oints of the plane and the p oints of a line on it by means of their radius-vectors. 1. Vectorial parametric equation of a line on a plane. Let;s consider a line on a plane with some coordinate system O, e1 , e2 . Let X b e some arbitrary p oint on this line (see Fig. 3.1) and let A b e some fixed p oint of this line. The p osition of the - - p oint X relative to the p oint A is marked by the vector AX , while the p osition of the p oint A itself is determined by its radius - - vector r0 = OA . Therefore we have - - - - r = OX = r0 + AX . (3.1)

Let's choose and fix some nonzero vector a = 0 directed along - - the line in question. The vector AX is expressed through a by means of the following formula: - - AX = a · t. (3.2)


§ 3. A STRAIGHT LINE ON A PLANE.

143

From the formulas (3.1) and (3.2) we immediately derive: r = r0 + a · t. (3.3)

Definition 3.1. The equality (3.3) is called the vectorial parametric equation of a line on a plane. The constant vector a = 0 in it is a directional vector of the line, while the variable t is a parameter. The constant vector r0 in (3.3) is the radius vector of an initial point. Each particular value of the parameter t corresp onds to some definite p oint on the line. The initial p oint A with the radius vector r0 is associated with the value t = 0. 2. Coordinate parametric equations of a line on a plane. Let's determine the vectors r, r0 , and a in the vectorial parametric equation (3.3) through their coordinates: r= x , y r0 = x0 , y0 a= ax . ay (3.4)

Due to (3.4) the equation (3.3) is written as two equations: x = x 0 + ax t , y = y 0 + ay t . (3.5)

Definition 3.2. The equalities (3.5) are called the coordinate parametric equations of a straight line on a plane. The constants ax and ay in them cannot vanish simultaneously. 3. Normal vectorial equation of a line on a plane. Let n = 0 b e a vector lying on the plane in question and b eing p erp endicular to the line in question (see Fig. 3.1). Let's apply the scalar multiplication by the vector n to b oth sides of the equation (3.3). As a result we get (r, n) = (r0 , n) + (a, n) t. (3.6)


144

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

But a n. Fort this reason the second term in the right hand side of (3.6) vanishes and the equation actually does not have the parameter t. The resulting equation is usually written as (r - r0 , n) = 0. (3.7)

The scalar product of two constant vectors r0 and n is a numeric constant. If we denote D = (r0 , n), then the equation (3.7) can b e written in the following form: (r, n) = D . (3.8)

Definition 3.3. Any one of the two equations (3.7) and (3.8) is called the normal vectorial equation of a line on a plane. The constant vector n = 0 in these equations is called a normal vector of this line. 4. Canonical equation of a line on a plane. Let's consider the case where ax = 0 and ay = 0 in the equations (3.5). In this case the parameter t can b e expressed through x and y : t= x - x0 , ax t= y - y0 . ay (3.9)

From the equations (3.9) we derive the following equality: x - x0 y - y0 = . ax ay If ax = 0, then the the first of the equations (3.5) turns to x = x0 . If ay = 0, then the second equation (3.5) turns to y = y0 . (3.12) (3.11) (3.10)


§ 3. A STRAIGHT LINE ON A PLANE.

145

Like the equation (3.10), the equations (3.11) and (3.12) do not comprise the parameter t. Definition 3.4. Any one of the three equalities (3.10), (3.11), and (3.12) is called the canonical equation of a line on a plane. The constants ax and ay in the equation (3.10) should b e nonzero. 5. The equation of a line passing through two given p oints on a plane. Assume that two distinct p oints A = B on a plane are given. We write their coordinates A = A(x0 , y0 ), B = B (x1 , y1 ). (3.13)

- - The vector a = AB can b e used for the directional vector of the line passing through the p oints A and B in (3.13). Then from (3.13) we derive the coordinates of a: a= ax ay = x1 - x0 . y1 - y0 (3.14)

Due to (3.14) the equation (3.10) can b e written as x - x0 y - y0 = . x1 - x0 y1 - y0 (3.15)

The equation (3.15) corresp onds to the case where x1 = x0 and y1 = y0 . If x1 = x0 , then we write the equation (3.11): x = x0 = x1 . If y1 = y0 , we write the equation (3.12): y = y0 = y1 . (3.17) (3.16)

The conditions x1 = x0 and y1 = y0 cannot b e fulfilled simultaneously since the p oints A and B are distinct, i. e. A = B .


146

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

Definition 3.5. Any one of the three equalities (3.15), (3.16), and (3.17) is called the equation of a line passing through two given points (3.13) on a plane. 6. General equation of a line on a plane. Let's apply the formula (29.8) from Chapter I in order to calculate the scalar product in (3.8). In this particular case it is written as
2 2

(r, n) =
i =1 j =1

r i n j gij ,

(3.18)

where gij are e1 , e2 on our drawn to b e skew-angular

the comp onents of the Gram matrix for the basis plane (see Fig. 3.1). In Fig. 3.1 the basis e1 , e2 is rectangular. However, in general case it could b e as well. Let's introduce the notations
2

ni =
j =1

n j gij .

(3.19)

The quantities n1 and n2 in (3.18) and in (3.19) are the coordinates of the normal vector n (see Fig. 3.1). Definition 3.6. The quantities n1 and n2 produced from the coordinates of the normal vector n by mens of the formula (3.19) are called the covariant coordinates of the vector n. Taking into account the notations (3.19), the formula (3.18) is written in the following simplified form:
2

(r, n) =
i =1

r i ni .

(3.20)

Let's recall the previous notations (3.4) and introduce new ones: A = n1 , B = n2 . (3.21)


§ 3. A STRAIGHT LINE ON A PLANE.

147

Due to (3.4), (3.20), and (3.21) the equation (3.8) is written as A x + B y - D = 0. (3.22)

Definition 3.7. The equation (3.22) is called the general equation of a line on a plane. 7. Double intersect equation of a line on a plane. Let's consider a line on a plane that does not pass through the origin and intersects with b oth of the coordinate axes. These conditions mean that D = 0, A = 0, and B = 0 in the equation (3.22) of this line. Through X and Y in Fig. 3.2 two intercept p oints are denoted: X = X (a, 0), Y = Y (0, b). (3.23)

The quantities a and b in (3.23) are expressed through the constant parameters A, B , and D of the equation (3.22) by means of the following formulas: a = D /A, b = D /B . (3.24)

The equation (3.22) itself in our present case can b e written as x y + = 1. D /A D /B If we take into account (3.24), the equality (3.25) turns to xy + = 1. a b
CopyRight c Sharipov R.A., 2010.

(3.25)

(3.26)


148

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

Definition 3.8. The equality (3.26) is called the double intercept equation of a line on a plane. The name of the equation (3.26) is due to the parameters a and b b eing the lengths of the segments [OX ] and [OY ] which the line intercepts on the coordinate axes. § 4. A plane in the space. Assume that some plane in the space E is chosen and fixed. In order to study various equations determining this plane we choose some coordinate system O, e1 , e2 , e3 in the space E. Then we can describ e the p oints of this plane by their radius vectors. 1. Vectorial parametric equation of a plane in the space. Let's denote through A some fixed p oint of the plane (see Fig. 4.1) and denote through X some arbitrary p oint of this plane. The p osition of the p oint X relative to the p oint A is marked - - by the vector AX , while the p osition of the p oint A is deter- - mined by its radius vector r0 = OA . For this reason - - - - r = OX = r0 + AX . Let's choose and fix some pair of non-collinear vector lying on the plane . Such vectors constitute a basis plane (see Definition 17.1 from Chapter I). The vector expanded in the basis of the vectors a and b: - - AX = a · t + b · . (4.1) sab on this - - AX is

(4.2)


§ 4. A PLANE IN THE SPACE.

149

Since X is an arbitrary p oint of the plane , the numb ers t and in (4.2) are two variable parameters. Up on substituting (4.2) into the formula (4.1), we get the following equality: r = r0 + a · t + b · . Definition 4.1. The equality (4.3) is called the parametric equation of a plane in the space. The non vectors a and b in it are called directional vectors of while t and are called parameters. The fixed vector radius vector of an initial point. (4.3) vectorial -collinear a plane, r0 is the

2. Coordinate parametric equation of a plane in the space. Let's determine the vectors r, r0 , a, b, in the vectorial parametric equation (4.3) through their coordinates: x y, z x0 y0 , z0 ax ay , az bx by . bz

r=

r0 =

a=

b=

(4.4)

Definition 4.2. The equalities (4.5) are called the coordinate parametric equations of a plane in the space. The triples of constants ax , ay , az and bx , by , bz in these equations cannot b e prop ortional to each other. 3. Normal vectorial equation of a plane in the space. Let n = 0 b e a vector p erp endicular to the plane (see Fig. 4.1). Let's apply the scalar multiplication by the vector n to b oth sides of the equation (4.3). As a result we get (r, n) = (r0 , n) + (a, n) t + (b, n) . (4.6)

Due to (4.4) the equation (4.3) is written as three equations x = x 0 + a x t + bx , y = y0 + ay t + by , (4.5) z = z0 + az t + bz .


150

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

But a terms in actually equation

n and b n. For this reason the second and the third the right hand side of (4.6) vanish and the equation does not have the parameters t and . The resulting is usually written as (r - r0 , n) = 0. (4.7)

The scalar product of two constant vectors r0 and n is a numeric constant. If we denote D = (r0 , n), then the equation (4.7) can b e written in the following form: (r, n) = D . Definitio is called the The constant vector of this (4.8)

n 4.3. Any one of the two equalities (4.7) and (4.8) normal vectorial equation of a plane in the space. vector n = 0 in these equations is called a normal plane. e space. The vector lying on the plane (4.7). Substituting relationship (4.9)

4. Canonical equation of a plane in th product of two non-coplanar vectors a b can b e chosen for the normal vector n in n = [a, b] into the equation (4.7), we get the (r - r0 , [a, b]) = 0.

Applying the definition of the mixed product (see formula (45.1) in Chapter I), the scalar product in the formula (4.9) can b e transformed to a mixed product: (r - r0 , a, b) = 0. Let's transform the equation (4.10) this purp ose we use the coordinate pr r0 , a, and b taken from (4.4). Apply Chapter I) and taking into account (4.10)

into a coordinate form. For esentations of the vectors r, ing the formula (50.8) from the fact that the oriented


§ 4. A PLANE IN THE SPACE.

151

volume of a basis (e1 , e2 , e3 ) is nonzero, we derive x-x ax b
x 0

y-y ay b
y

0

z - z0 az = 0. b
z

(4.11)

If we use not the equation (4.7), but the equation (4.8), then instead of the equation (4.10) we get the following relationship: (r, a, b) = D . In the coordinate form the relationship (4.12) is written as x ax b
x

(4.12)

y ay b
y

z az = D . bz

(4.13)

Definition 4.4. Any one of the two equalities (4.11) and (4.13) is called the canonical equation of a plane in the space. The triples of constants ax , ay , az and bx , by , bz in these equations should not b e prop ortional to each other. Definition 4.5. The equation (4.10) and the equation (4.12), where a b, are called the vectorial forms of the canonical equation of a plane in the space. 5. The equation of a plane passing through three given p oints. Assume that three p oints A, B , and C in the space E nit lying on a straight line are given. We write their coordinates A = A(x0 , y0 , z0 ), B = B (x1 , y1 , z1 ), C = C (x2 , y2 , z2 ). - - - - The vectors a = AB and b = AC can b e chosen for the directional vectors of a plane passing through the p oints A, B , (4.14)


152

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

and C . Then from the formulas (4.14) we derive ax ay az x1 - x0 = y1 - y0 , z1 - z0 bx by bz x2 - x0 = y2 - y0 . z2 - z0

a=

b=

(4.15)

Due to (4.15) the equation (4.11) can b e written as x - x0 x1 - x0 x2 - x
0

y - y0 y1 - y0 y2 - y
0

z - z0 z1 - z0 = 0. z2 - z0

(4.16)

If we denote through r1 and r2 the radius vectors of the p oints B and C from (4.14), then (4.15) can b e written as a = r1 - r0 , b = r2 - r0 . (4.17)

Substituting (4.17) into the equation (4.10), we get (r - r0 , r1 - r0 , r2 - r0 ) = 0. (4.18)

Definition 4.6. The equality (4.16) is called the equation of a plane passing through three given points. Definition 4.7. The equality (4.18) is called the vectorial form of the equation of a plane passing through three given points. 6. General equation of a plane in the space. Let's apply the formula (29.8) from Chapter I in order to calculate the scalar product (r, n) in the equation (4.8). This yields
3 3

(r, n) =
i =1 j =1

r i n j gij ,

(4.19)

where gij are the comp onents of the Gram matrix for the basis e1 , e2 , e3 (see Fig. 4.1). In Fig. 4.1 the basis e1 , e2 , e3 is


§ 4. A PLANE IN THE SPACE.

153

drawn to b e rectangular. However, in general case it could b e skew-angular as well. Let's introduce the notations
3

ni =
j =1

n j gij .

(4.20)

The quantities n1 , n2 , and n3 in (4.19) and (4.20) are the coordinates of the normal vector n (see Fig. 4.1). Definition 4.8. The quantities n1 , n2 , and n3 produced from the coordinates of the normal vector n by mens of the formula (4.20) are called the covariant coordinates of the vector n. Taking into account the notations (4.20), the formula (4.19) is written in the following simplified form:
2

(r, n) =
i =1

r i ni .

(4.21)

Let's recall the previous notations (4.4) and introduce new ones: A = n1 , B = n2 , C = n3 . (4.22)

Due to (4.4), (4.21), and (4.22) the equation (4.8) is written as A x + B y + C y - D = 0. (4.23)

Definition 4.9. The equation (4.23) is called the general equation of a plane in the space. 7. Triple intercept equation of a plane in the space. Let's consider a plane in the space that does not pass through the origin and intersects with each of the three coordinate axes. These conditions mean that D = 0, A = 0, B = 0, and C = 0 in (4.23). Through X , Y , and Z in Fig. 4.2 b elow we denote three


154

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

intercept p oints of the plane. Here are the coordinates of these three intercept p oints X , Y , and Z produced by our plane: X = X (a, 0, 0), Y = Y (0, b, 0), Z = Y (0, 0, c). The quantities a, b, and c (4.24) are expressed through constant parameters A, B , and D of the equation (4.23) means of the formulas a = D /A, b = D /B , c = D /C. The equation of the plane (4.23) itself can b e written as y z x + + = 1. D /A D /B D /C If we take into account (4.25) , the equality (4.26) turns to z xy + + = 1. a b c (4.27) (4.26) (4.25) in the C, by (4.24)

Definition 4.10. The equality (4.27) is called the triple intercept equation of a plane in the space. § 5. A straight line in the space. Assume that some straight line a in the space E is chosen and fixed. In order to study various equations determining this line we choose some coordinate system O, e1 , e2 , e3 in the space E. Then we can describ e the p oints of the line by their radius
CopyRight c Sharipov R.A., 2010.


§ 5. A STRAIGHT LINE IN THE SPACE.

155

vectors relative to the origin O. 1. Vectorial parametric equation of a line in the space. Let's denote through A some fixed p oint on the line (see Fig. 5.1) and denote through X an arbitrary p oint of this line. The p osition of the p oint X relative to the p oint A is marked by the - - vector AX , while the p osition of the p oint A itself is determined - - by its radius vector r0 = OA . Therefore we have - - r = r0 + AX . (5.1)

Let's choose and fix some nonzero vector a = 0 directed along the line in question. The vector - - AX is expressed through the vector a by means of the formula - - AX = a · t. From the formulas (5.1) and (5.2) we immediately derive r = r0 + a · t. (5.3) (5.2)

Definition 5.1. The equality (5.3) is called the vectorial parametric equation of the line in the space. The constant vector a = 0 in this equation is called a directional vector of the line, while t is a parameter. The constant vector r0 is the radius vector of an initial point. Each particular value of the parameter t corresp onds to some definite p oint on the line. The initial p oint A with the radius vector r0 is associated with the value t = 0.


156

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

2. Coordinate parametric equations of a line in the space. Let's determine the vectors r, r0 , and a in the vectorial parametric equation (5.3) through their coordinates: x r= y , z x0 y0 , z0 ax ay . az

r0 =

a=

(5.4)

Definition 5.2. The equalities (5.5) are called the coordinate parametric equations of a line in the space. The constants ax , ay , az in them cannot vanish simultaneously 3. Vectorial equation of a line in the space. Let's apply the vectorial multiplication by the vector a to b oth sides of the equation (5.3). As a result we get [r, a] = [r0 , a] + [a, a] t. (5.6)

Due to (5.4) the equation x y z

(5.3) is written as three equations: = x0 + ax t, = y 0 + ay t , = z0 + az t. (5.5)

Due to the item 4 of the theorem 39.1 in Chapter I the vector product [a, a] in (5.6) is equal to zero. For this reason the equation (5.6) actually does not contain the parameter t. This equation is usually written as follows: [r - r0 , a] = 0. (5.7)

The vector product of two constant vectors r0 and a is a constant vector. If we denote this vector b = [r0 , a], then the equation of the line (5.7) can b e written as [r, a] = b, where b a. (5.8)


§ 5. A STRAIGHT LINE IN THE SPACE.

157

Definition 5.3. Any one of the two equalities (5.7) and (5.8) is called the vectorial 1 equation of a line in the space. The constant vector b in the equation (5.8) should b e p erp endicular to the directional vector a. 4. Canonical equation of a line in the space. Let's consider the case where ax = 0, ay = 0, and az = 0 in the equations (5.5). Then the parameter t can b e expressed through x, y , and z : t= x - x0 , ax t= y - y0 , ay t= z - z0 . az (5.9)

From the equations (5.9) one can derive the equalities y - y0 z - z0 x - x0 = = . ax ay az (5.10)

If ax = 0, then instead of the first equation (5.9) from (5.5) we derive x = x0 . Therefore instead of (5.10) we write x = x0 , y - y0 z - z0 = . ay az (5.11)

If ay = 0, then instead of the second equation (5.9) from (5.5) we derive y = y0 . Therefore instead of (5.10) we write y = y0 , x - x0 z - z0 = . ax az (5.12)

If az = 0, then instead of the third equation (5.9) from (5.5) we derivez = z0 . Therefore instead of (5.10) we write z = z0 , x - x0 y - y0 = . ax ay (5.13)

1 The terms « vectorial equation» and « vectorial parametric equation» are often confused and the term « vector equation» is misapplied to (5.3).


158

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

If ax = 0 and ay = 0, then instead of (5.10) we write x = x0 , y = y0 . (5.14)

If ax = 0 and az = 0, then instead of (5.10) we write x = x0 , z = z0 . (5.15)

If ay = 0 and az = 0, then instead of (5.10) we write y = y0 , z = z0 . (5.16)

Definition 5.4. Any one of the seven pairs of equations (5.10), (5.11) , (5.12) , (5.13) , (5.14) , (5.15) , and (5.16) is called the canonical equation of a line in the space. 5. The equation of a line passing through two given p oints in the space. Assume that two distinct p oints A = B in the space are given. We write their coordinates A = A(x0 , y0 , z0 ), B = B (x1 , y1 , z1 ). (5.17)

- - The vector a = AB can b e used for the directional vector of the line passing through the p oints A and B in (5.17). Then from (5.17) we derive the coordinates of a: ax a = ay az x1 - x0 y1 - y0 . z1 - z0

=

(5.18)

Due to (5.18) the equations (5.10) can b e written as x - x0 y - y0 z - z0 = = . x1 - x0 y1 - y0 z1 - z0 (5.19)


§ 5. A STRAIGHT LINE IN THE SPACE.

159

The equations (5.19) corresp ond to the case where the inequalities x1 = x0 , y1 = y0 , and z1 = z0 are fulfilled. If x1 = x0 , then instead of (5.19) we write the equations x = x0 = x1 , z - z0 y - y0 = . y1 - y0 z1 - z0 (5.20)

If y1 = y0 , then instead of (5.19) we write the equations y = y0 = y1 , z - z0 x - x0 = . x1 - x0 z1 - z0 (5.21)

If z1 = z0 , then instead of (5.19) we write the equations z = z0 = z1 , x - x0 y - y0 = . x1 - x0 y1 - y0 (5.22)

If x1 = x0 and y1 = y0 , then instead of (5.19) we write x = x0 = x1 , y = y0 = y1 . (5.23)

If x1 = x0 and z1 = z0 , then instead of (5.19) we write x = x0 = x1 , z = z0 = z1 . (5.24)

If y1 = y0 and z1 = z0 , then instead of (5.19) we write y = y0 = y1 , z = z0 = z1 . (5.25)

The conditions x1 = x0 , y1 = y0 , and z1 = z0 cannot b e fulfilled simultaneously since A = B . Definition 5.5. Any one of the seven pairs of equalities (5.19), (5.20), (5.21), (5.22) , (5.23), (5.24),and (5.25) is called the equation of a line passing through two given points A and B with the coordinates (5.17).


160

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

6. The equation of a line in the space as the intersection of two planes. In the vectorial form the equations of two intersecting planes can b e written as (4.8): (r, n1 ) = D1 , (r, n2 ) = D2 . (5.26)

For the planes given by the equations (5.26) do actually intersect their normal vectors should b e non-parallel: n1 n2 . In the coordinate form the equations of two intersecting planes can b e written as (4.23): A1 x + B1 y + C1 y - D1 = 0, A2 x + B2 y + C2 y - D2 = 0. (5.27)

Definition 5.6. Any one of the two pairs of equalities (5.26) and (5.27) is called the equation of a line in the space obtained as the intersection of two planes. § 6. Ellipse. Canonical equation of an ellipse. Definition 6.1. An el lipse is a set of p oints on some plane the sum of distances from each of which to some fixed p oints F1 and F2 of this plane is a constant which is the same for all p oints of the set. The p oints F1 and F2 are called the foci of the ellipse. Assume that an ellipse with the foci F1 and F2 is given. Let's draw the line connecting the p oints F1 and F2 and choose this line for the x-axis of a coordinate system. Let's denote through O the midp oint of the segment [F1 F2 ] and choose it for the origin. We choose the second coordinate axis (the y -axis) to


§ 6. ELLIPSE.

161

b e p erp endicular to the x-axis on the ellipse plane (see Fig. 6.1). We choose the unity scales along the axes. This means that the basis of the coordinate system constructed is orthonormal. Let M = M (x, y ) b e some arbitrary p oint of the ellipse. According to the definition 6.1 the sum |M F1 | + |M F2 | is equal to a constant which does not dep end on the p osition of M on the ellipse. Let's denoter this constant through 2 a and write |M F1 | + |M F2 | = 2 a. (6.1)

The length of the segment [F1 F2 ] is also a constant. Let's denote this constant through 2 c. This yields the relationships |F1 O| = |OF2 | = c, |F1 F2 | = 2 c. (6.2)

From the triangle inequality |F1 F2 | |M F1 | + |M F2 | we derive the following inequality for the constants c and a: c a. (6.3)

The case c = a in (6.3) corresp onds to a degenerate ellipse. In this case the triangle inequality |F1 F2 | |M F1 | + |M F2 | turns to the equality |F1 F2 | = |M F1 | + |M F2 |, while the M F1 F2 itself collapses to the segment [F1 F2 ]. Since M [F1 F2 ], a degenerate ellipse with the foci F1 and F2 is the segment [F1 F2 ]. The case of a degenerate ellipse is usually excluded by setting 0 c < a. (6.4)

The formulas (6.2) determine the foci F1 and F2 of the ellipse in the coordinate system we have chosen: F1 = F1 (-c, 0), F2 = F2 (c, 0). (6.5)

Having determined the coordinates of the p oints F1 and F2 and knowing the coordinates of the p oint M = M (x, y ), we derive the
CopyRight c Sharipov R.A., 2010.


162

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

following relationships for the segments [M F1 ] and [M F2 ]: |M F1 | = |M F2 | = y 2 + (x + c)2 , y 2 + (x - c)2 . (6.6)

Derivation of the canonical equation of an ellipse. Let's substitute (6.6) into the equality (6.1) defining the ellipse: y 2 + (x + c)2 + y 2 + (x - c)2 = 2 a. (6.7)

Then we move one of the square roots to the right hand side of the formula (6.7). As a result we derive y 2 + (x + c)2 = 2 a - y 2 + (x - c)2 . (6.8)

Let's square b oth sides of the equality (6.8): y 2 + (x + c)2 = 4 a2 - -4a (6.9)

y 2 + (x - c)2 + y 2 + (x - c)2 .

Up on expanding brackets and collecting similar terms the equality (6.9) can b e written in the following form: 4a y 2 + (x - c)2 = 4 a2 - 4 x c. (6.10)

Let's cancel the factor four in (6.10) and square b oth sides of this equality. As a result we get the formula a2 (y 2 + (x - c)2 ) = a4 - 2 a2 x c + x2 c2 . (6.11)

Up on expanding brackets and recollecting similar terms the equality (6.11) can b e written in the following form: x2 (a2 - c2 ) + y 2 a2 = a2 (a2 - c2 ). (6.12)


§ 6. ELLIPSE.

163

Both sides of the equality (6.12) contain the quantity a2 - c2 . Due to the inequality (6.4) this quantity is p ositive. For this reason it can b e written as the square of some numb er b > 0: a2 - c2 = b2 . Due to (6.13) the equality (6.12) can b e written as x2 b2 + y 2 a2 = a2 b2 . (6.14) (6.13)

Since b > 0 and a > 0 (see the inequalities (6.4)), the equality (6.14) transforms to the following one: x2 y2 + 2 = 1. a2 b (6.15)

Definition 6.2. The equality (6.15) is called the canonical equation of an ellipse. Theorem 6.1. For each p oint M = M (x, y ) of the ellipse determined by the initial equation (6.7) its coordinates ob ey the canonical equation (6.15). The proof of the theorem 6.1 consists in the ab ove calculations that lead to the canonical equation of an ellipse (6.15). This canonical equation yields the following imp ortant inequalities: |x| a, |y | b. (6.16)

Theorem 6.2. The canonical equation of an ellipse (6.15) is equivalent to the initial equation (6.7). Proof. In order to prove the equation 6.2 we transform the expressions (6.6) relying on (6.15) . From (6.15) we derive y 2 = b2 - b2 2 x. a2 (6.17)


164

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

Substituting (6.17) into the first formula (6.6), we get |M F1 | = = b2 - b2 2 x + x2 + 2 x c + c2 = a2

a 2 - b2 2 x + 2 x c + (b2 + c2 ). a2

(6.18)

Now we take into account the relationship (6.13) and write the equality (6.18) in the following form: |M F1 | = a2 + 2 x c + c2 2 x= a2 a2 + c x a
2

.

(6.19)

Up on calculating the square root the formula (6.19) yields |M F1 | = |a2 + c x| . a (6.20)

From the inequalities (6.4) and (6.16) we derive a2 + c x > 0. Therefore the modulus signs in (6.20) can b e omitted: |M F1 | = cx a2 + c x =a+ . a a (6.21)

In the case of the second formula (6.6) the considerations similar to the ab ove ones yield the following result: |M F2 | = cx a2 - c x =a- . a a (6.22)

Now it is easy to see that the equation (6.7) written as |M F1 | + |M F2 | = 2 a due to (6.6) is immediate from (6.21) and (6.22). The theorem 6.2 is proved. Let's consider again the inequalities (6.16). The coordinates of any p oint M of the ellipse ob ey these inequalities. The first of the inequalities (6.16) turns to an equality if M coincides with A or if M coincides with C (see Fig. 6.1). The second inequality (6.16) turns to an equality if M coincides with B or if M coincides D .


§ 7 . T H E E C C E N T R I C I T Y A N D DI R E C T R I C E S . . .

165

Definition 6.3. The p oints A, B , C , and D in Fig. 6.1 are called the vertices of an ellipse. The segments [AC ] and [B D ] are called the axes of an ellipse, while the segments [OA], [OB ], [OC ], and [OD ] are called its semiaxes. The constants a, b, and c ob ey the relationship (6.13). From this relationship and from the inequalities (6.4) we derive 0
Definition 6.4. Due to the inequalities (6.23) the semiaxis [OA] in Fig. 6.1 is called the major semiaxis of the ellipse, while the semiaxis [OB ] is called its minor semiaxis. Definition 6.5. A coordinate system O, e1 , e2 with an orthonormal basis e1 , e2 where an ellipse is given by its canonical equation (6.15) and where the inequalities (6.23) are fulfilled is called a canonical coordinate system of this ellipse. § 7. The eccentricity and directrices of an ellipse. The prop erty of directrices. The shap e and sizes of an ellipse are determined by two constants a and b in its canonical equation (6.15). Due to the relationship (6.13) the constant b can b e expressed through the constant c. Multiplying b oth constants a and c by the same numb er, we change the sizes of an ellipse, but do not change its shap e. The ratio of these two constants = c . a (7.1)

is resp onsible for the shap e of an ellipse. Definition 7.1. The quantity defined by the relationship (7.1), where a is the ma jor semiaxis and c is the half of the interfocal distance, is called the eccentricity of an ellipse.


166

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

The eccentricity (7.1) is used in order to define one more numeric parameter of an ellipse. It is usually denoted through d: d= a2 a =. c (7.2)

Definition 7.2. On the plane of an ellipse there are two lines parallel to its minor axis and placed at the distance d given by the formula (7.2) from its center. These lines are called directrices of an ellipse. Each ellipse has two foci and two directrices. Each directrix has the corresp onding focus of it. This is that of two foci which is more close to the directrix in question. Let M = M (x, y ) b e some arbitrary p oint of an ellipse. Let's connect this p oint with the left focus of the ellipse F1 and drop the p erp endicular from it to the left directrix of the ellipse. Let's denote through H1 the base of such a p erp endicular and calculate its length: |M H1 | = |x - (-d)| = |d + x|. (7.3)

Taking into account (7.2), the formula (7.3) can b e brought to |M H1 | = |a2 + c x| a2 +x = . c c (7.4)

The length of the segment M F1 was already calculated ab ove. Initially it was given by one of the formulas (6.6), but later the more simple expression (6.20) was derived for it: |M F1 | = |a2 + c x| . a (7.5)


§ 8. THE EQUATION OF A TANGENT LINE . . .

167

If we divide (7.5) by (7.4), we obtain the following relationship: |M F1 | c = = . |M H1 | a (7.6)

The p oint M can change its p osition on the ellipse. Then the numerator and the denominator of the fraction (7.6) are changed, but its value remains unchanged. This fact is known as the prop erty of directrices. Theorem 7.1. The ratio of the distance from some arbitrary p oint M of an ellipse to its focus and the distance from this p oint to the corresp onding directrix is a constant which is equal to the eccentricity of the ellipse. § 8. The equation of a tangent line to an ellipse. Let's consider an ellipse given by its canonical equation (6.15) in its canonical coordinate system (see Definition 6.5). Let's draw a tangent line to this ellipse and denote through M = M (x0 , y0 ) its tangency p oint (see Fig. 8.1). Our goal is to write the equation of a tangent line to an ellipse. An ellipse is a curve comp osed by two halves -- the upp er half and the lower half. Any one of these two halves can b e treated as a graph of a function y = f (x) (8.1)

with the domain (-a, a). The equation of a tangent line to the graph of a function is given by the following well-known formula (see [9]): y = y0 + f (x0 ) (x - x0 ). (8.2)


168

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

In order to apply the formula (8.2) to an ellipse we need to calculate the derivative of the function (8.1). Let's substitute the expression (8.1) for y into the formula (6.15): x2 (f (x))2 + = 1. a2 b2 (8.3)

The equality (8.3) is fulfilled identically in x. Let's differentiate the equality (8.3) with resp ect to x. This yields 2 x 2 f (x) f (x) + = 0. a2 b2 (8.4)

Let's apply the formula (8.4) for to calculate the derivative f (x): f (x) = - b2 x . a2 f (x) (8.5)

In order to substitute (8.5) into the equation (8.2) we change x for x0 and f (x) for f (x0 ) = y0 . As a result we get f (x0 ) = - b2 x 0 . a2 y 0 (8.6)

Let's substitute (8.6) into the equation of the tangent line (8.2). This yields the following relationship y = y0 - b2 x 0 (x - x0 ). a2 y 0 (8.7)

Eliminating the denominator, we write the equality (8.7) as
2 a2 y y0 + b2 x x0 = a2 y0 + b2 x2 . 0

(8.8)

Now let's divide b oth sides of the equality (8.8) by a2 b2 :
2 y y0 x2 y0 x x0 0 + 2 = 2 + 2. a2 b a b

(8.9)

CopyRight c Sharipov R.A., 2010.


§ 8. THE EQUATION OF A TANGENT LINE . . .

169

Note that the p oint M = M (x0 , y9 ) is on the ellipse. Therefore its coordinates satisfy the equation (6.15):
2 x2 y0 0 + 2 = 1. a2 b

(8.10)

Taking into account (8.10), we can transform (8.9) to x x0 y y0 + 2 = 1. a2 b (8.11)

This is the required equation of a tangent line to an ellipse. Theorem 8.1. For an ellipse determined by its canonical equation (6.15) its tangent line that touches this ellipse at the p oint M = M (x0 , y0 ) is given by the equation (8.11). The equation (8.11) is a particular case of the equation (3.22) where the constants A, B , and D are given by the formulas A= x0 , a2 B= y0 , b2 D = 1. (8.12)

According to the definition 3.6 and the formulas (3.21) the constants A and B in (8.12) are the covariant comp onents of the normal vector for the tangent line to an ellipse. The tangent line equation (8.11) is written in a canonical coordinate system of an ellipse. The basis of such a coordinate system is orthonormal. Therefore the formula (3.19) and the formula (32.4) from Chapter I yield the following relationships: A = n1 = n1 , B = n2 = n2 . (8.13)

Theorem 8.2. The quantities A and B in (8.12) are the coordinates of the normal vector n for the tangent line to an ellipse which is given by the equation (8.11). The relationships (8.13) prove the theorem 8.2.


170

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

§ 9. Focal prop erty of an ellipse. The term focus is well known in optics. It means a p oint where light rays converge up on refracting in lenses or up on reflecting in curved mirrors. In the case of an ellipse let's assume that it is manufactured of a thin strip of some flexible material and assume that its inner surface is covered with a light reflecting layer. For such a materialized ellipse one can formulate the following focal prop erty. Theorem 9.1. A light ray emitted from one of the foci of an ellipse up on reflecting on its inner surface passes through the other focus of this ellipse. Theorem 9.2. The p erp endicular to a tangent line of an ellipse drawn at the tangency p oint is a bisector in the triangle comp osed by the tangency p oint and two foci of the ellipse. The theorem 9.2 is a geometric version of the theorem 9.1. These theorems are equivalent due to the reflection law saying that the angle of reflection is equal to the angle of incidence. Proof of the theorem 9.2. Let's choose an arbitrary p oint M = M (x0 , y0 ) on the ellipse and draw the tangent line to the ellipse through this p oint as shown in Fig. 9.1. Then we draw the p erp endicular [M N ] to the tangent line through the p oint M . The segment [M N ] is directed along the normal vector of the tangent line. In order to prove that this segment is a bisector of the triangle F1 M F2 it is sufficient to prove the equality cos(F1 M N ) = cos(F2 M N ). (9.1)

The cosine equality (9.1) is equivalent to the following equality


§ 9. FOCAL PROPERTY OF AN ELLIPSE.

171

for the scalar products of vectors: -- - -- - (M F2 , n) (M F1 , n) = . |M F1 | |M F2 | The coordinates ordinate system of The coordinates of Therefore we can fi -- - M F2 used in the ab -- - M F1 = (9.2)

of the p oints F1 and F2 in a canonical cothe ellipse are known (see formulas (6.5)). the p oint M = M (x0 , y0 ) are also known. -- - nd the coordinates of the vectors M F1 and ove formula (9.2):
0

-c - x -y0

,

-- - c-x M F2 = -y0

0

.

(9.3)

The coordinates of the normal vector n of the tangent line in Fig. 9.1 are given by the formulas (8.12) and (8.13): x0 a2 y0 b2

n=

.

(9.4)

Using (9.3) and (9.4), we apply the formula (33.3) from Chapter I for calculating the scalar products in (9.2):
2 2 y0 - c x0 x2 y0 - c x0 - x2 -- - 0 0 - 2= - 2 - 2, (M F1 , n) = a2 b a2 a b 2 2 c x0 - x2 y0 c x0 x2 y0 -- - 0 0 (M F2 , n) = - 2 = 2 - 2 - 2. a2 b a a b

(9.5)

The p oint M = M (x0 , y0 ) lies on the ellipse. Therefore its coordinates should satisfy the equation of the ellipse (6.15):
2 y0 x2 0 + 2 = 1. a2 b

(9.6)


172

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

Due to (9.6) the formulas (9.5) simplify and take the form a2 + c x 0 - c x0 -- - -1=- , (M F1 , n) = a2 a2 a2 - c x 0 c x0 -- - . (M F2 , n) = 2 - 1 = - a a2

(9.7)

In order to calculate the denominators in the formula (9.2) we use the formulas (6.21) and (6.22). In this case, when applied to the p oint M = M (x0 , y0 ), they yield |M F1 | = a2 + c x 0 , a |M F2 | = a2 - c x0 . a (9.8)

From the formulas (9.7) and (9.8) we easily derive the equalities -- - 1 (M F1 , n) =- , |M F1 | a -- - (M F2 , n) 1 =- |M F2 | a

which proves the equality (9.2). As a result the theorem 9.2 and the theorem 9.1 equivalent to it b oth are proved. § 10. Hyp erb ola. Canonical equation of a hyp erb ola. Definition 10.1. A hyperbola is a set of p oints on some plane the modulus of the difference of distances from each of which to some fixed p oints F1 and F2 of this plane is a constant which is the same for all p oints of the set. The p oints F1 and F2 are called the foci of the hyp erb ola. Assume that a hyp erb ola with the foci F1 and F2 is given. Let's draw the line connecting the p oints F1 and F2 and choose this line for the x-axis of a coordinate system. Let's denote through O the midp oint of the segment [F1 F2 ] and choose it for the origin. We choose the second coordinate axis (the y axis) to b e p erp endicular to the x-axis on the hyp erb ola plane


§ 10. HYPERBOLA.

173

(see Fig. 10.1). We choose the unity scales along the axes. This means that the basis of the coordinate system constructed is orthonormal. Let M = M (x, y ) b e some arbitrary p oint of the hyp erb ola in Fig. 10.1. According to the definition of a hyp erb ola 10.1 the modulus of the difference |M F1 | - |M F2 | is a constant which does not dep end on a location of the p oint M on the hyp erb ola. We denote this constant through 2 a and write the formula |M F1 | - |M F2 | = 2 a. (10.1)

According to (10.1) the p oints of the hyp erb ola are sub divided into two subsets. For the p oints of one of these two subsets the condition (10.1) is written as |M F1 | - |M F2 | = 2 a. (10.2)

These p oints constitute the right branch of the hyp erb ola. For the p oints of the other subset the condition (10.1) is written as |M F2 | - |M F1 | = 2 a. (10.3)

These p oints constitute the left branch of the hyp erb ola. The length of the segment [F1 F2 ] is also a constant. Let's denote this constant through 2 c. This yields |F1 O| = |OF2 | = c, From the triangle inequality |F1 F2 | |F1 F2 | = 2 c. (10.4)

|M F1 | - |M F2 | , from


174

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

(10.2), from (10.3) , and from (10.4) we derive c Th e b ola. turns tinguis |F1 F2 | a. (10.5)

case c = a in (10.5) corresp onds to a degenerate hyp erIn this case the inequality |F1 F2 | |M F1 | - |M F2 | to the equality |F1 F2 | = |M F1 | - |M F2 | which dishes two sub cases. In the first sub case the equality = |M F1 | - |M F2 | is written as |F1 F2 | = |M F1 | - |M F2 |. (10.6)

The equality (10.6) means that the triangle F1 M F2 collapses into the segment [F1 M ], i. e. the p oint M lies on the ray going along the x-axis to the right from the p oint F2 (see Fig. 10.1). In the second sub case of the case c = a the equality |F1 F2 | = = |M F1 | - |M F2 | is written as follows |F1 F2 | = |M F2 | - |M F1 |. (10.7)

The equality (10.7) means that the triangle F1 M F2 collapses into the segment [F1 M ], i. e. the p oint M lies on the ray going along the x-axis to the left from the p oint F1 (see Fig. 10.1). As we see considering the ab ove two sub cases, if c = a the degenerate hyp erb ola is the union of two non-intersecting rays lying on the x-axis and b eing opp osite to each other. Another form of a degenerate hyp erb ola arises if a = 0. In this case the branches of the hyp erb ola straighten and, gluing with each other, lie on the y -axis. Both cases of degenerate hyp erb olas are usually excluded from the consideration by means of the following inequalities: c > a > 0. (10.8)

The formulas (10.4) determine the coordinates of the foci F1


§ 10. HYPERBOLA.

175

and F2 of our hyp erb ola in the chosen coordinate system: F1 = F1 (-c, 0), F2 = F2 (c, 0). (10.9)

Knowing the coordinates of the p oints F1 and F2 and knowing the coordinates of the p oint M = M (x, y ), we write the formulas |M F1 | = |M F2 | = y 2 + (x + c)2 , y 2 + (x - c)2 . (10.10)

Derivation of the canonical equation of a hyp erb ola. As we have seen ab ove, the equality (10.1) defining a hyp erb ola breaks into two equalities (10.2) and (10.3) corresp onding to the right and left branches of the hyp erb ola. Let's unite them back into a single equality of the form |M F1 | - |M F2 | = ± 2 a. (10.11)

Let's substitute the formulas (10.10) into the equality (10.11) : y 2 + (x + c)2 - y 2 + (x - c)2 = ± 2 a. (10.12)

Then we move one of the square roots to the right hand side of the formula (10.12) . As a result we derive y 2 + (x + c)2 = ± 2 a + y 2 + (x - c)2 . (10.13)

Squaring b oth sides of the equality (10.13), we get y 2 + (x + c)2 = 4 a2 ± ±4a (10.14)

y 2 + (x - c)2 + y 2 + (x - c)2 .

Up on expanding brackets and collecting similar terms the equality (10.14) can b e written in the following form: 4 a y 2 + (x - c)2 = 4 a2 - 4 x c. (10.15)

CopyRight c Sharipov R.A., 2010.


176

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

Let's cancel the factor four in (10.15) and square b oth sides of this equality. As a result we get the formula a2 (y 2 + (x - c)2 ) = a4 - 2 a2 x c + x2 c2 . (10.16)

Up on expanding brackets and recollecting similar terms the equality (10.16) can b e written in the following form: x2 (a2 - c2 ) + y 2 a2 = a2 (a2 - c2 ). (10.17)

An attentive reader can note that the ab ove calculations ale almost literally the same as the corresp onding calculations for the case of an ellipse. As for the resulting formula (10.17) , it coincides exactly with the formula (6.12) . But, nevertheless, there is a difference. It consists in the inequalities (10.8), which are different from the inequalities (6.4) for an ellipse. Both sides of the equality (10.17) comprises the quantity a2 - c2 . Due to the inequalities (10.8) this quantity is negative. For this reason the quantity a2 - c2 can b e written as the square of some p ositive quantity b > 0 taken with the minus sign: a2 - c2 = -b2 . Due to (10.18) the equality (10.17) can b e written as -x2 b2 + y 2 a2 = -a2 b2 . (10.19) (10.18)

Since b > 0 and a > 0 (see inequalities (10.8)), the ab ove equality (10.19) transforms to the following one: y2 x2 - 2 = 1. a2 b (10.20)

Definition 10.2. The equality (10.20) is called the canonical equation of a hyp erb ola.


§ 10. HYPERBOLA.

177

Theorem 10.1. For each p oint M (x, y ) of the hyp erb ola determined by the initial equation (10.12) its coordinates satisfy the canonical equation (10.20). The ab ove derivation of the canonical equation (10.20) of a hyp erb ola proves the theorem 10.1. The canonical equation leads (10.20) to the following imp ortant inequality: |x| a. (10.21)

Theorem 10.2. The canonical equation of a hyp erb ola (10.20) is equivalent to the initial equation (10.12) . Proof. The proof of the theorem 10.2 is analogous to the proof of the theorem 6.2. In order to prove the theorem 10.2 we calculate the expressions (10.10) relying on the equation (10.20). The equation (10.20) itself can b e written as follows: y2 = b2 2 x - b2 . a2 (10.22)

Substituting (10.22) into the first formula (10.10) , we get |M F1 | = = b2 2 x - b2 + x2 + 2 x c + c2 = a2 a2 + b2 2 x + 2 x c + (c2 - b2 ). a2

(10.23)

Now we take into account the relationship (10.18) and write the equality (10.23) in the following form: |M F1 | = a2 + 2 x c + c2 2 x= a2 a2 + c x a
2

.

(10.24)

Up on calculating the square root the formula (10.24) yields |M F1 | = |a2 + c x| . a (10.25)


178

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

From the inequalities (10.8) and (10.21) we derive |c x| > a2 . Therefore the formula (10.25) can b e written as follows: |M F1 | = c |x| c |x| + sign(x) a2 = + sign(x) a. a a (10.26)

In the case of the second formula (10.10) the considerations, analogous to the ab ove ones, yield the following result: |M F2 | = c |x| c |x| - sign(x) a2 = - sign(x) a. a a (10.27)

Let's subtract the equality (10.27) from the equality (10.26) . Then we get the following relationships: |M F1 | - |M F2 | = 2 sign(x) a = ± 2 a. (10.28)

The plus sign in (10.28) corresp onds to the case x > 0, which corresp onds to the right branch of the hyp erb ola in Fig. 10.1. The minus sign corresp onds to the left branch of the hyp erb ola. Due to what was said the equality (10.28) is equivalent to the equality (10.11), which in turn is equivalent to the equality (10.12) . The theorem 10.2 is proved. Let's consider again the inequality (10.21) which should b e fulfilled for the x-coordinate of any p oint M on the hyp erb ola. The inequality (10.21) turns to an equality if M coincides with A or if M coincides with C (see Fig. 10.1). Definition 10.3. The p oints A and C in Fig. 10.1 are called the vertices of the hyp erb ola. The segment [AC ] is called the transverse axis or the real axis of the hyp erb ola, while the segments [OA] and [OC ] are called its real semiaxes. The constant a in the equation of the hyp erb ola (10.20) is the length of the segment [OA] in Fig. 10.1 (the length of the real semiaxis of a hyp erb ola). As for the constant b, there is no


§ 1 1 . T H E E C C E N T R I C I T Y A N D DI R E C T R I C E S . . .

179

segment of the length b in Fig. 10.1. For this reason the constant b is called the length of the imaginary semiaxis of a hyp erb ola, i. e. the semiaxis which does not actually exist. Definition 10.4. A coordinate system O, e1 , e2 with an orthonormal basis e1 , e2 where a hyp erb ola is given by its canonical equation (10.20) is called a canonical coordinate system of this hyp erb ola. § 11. The eccentricity and directrices of a hyp erb ola. The prop erty of directrices. The shap e and sizes of a hyp erb ola are determined by two constants a and b in its canonical equation (10.20) .. Due to the relationship (10.18) the constant b can b e expressed through the constant c. Multiplying b oth constants a and c by the same numb er, we change the sizes of a hyp erb ola, but do not change its shap e. The ratio of these two constants = c . a (11.1)

is resp onsible for the shap e of a hyp erb ola. Definition 11.1. The quantity defined by the relationship (11.1), where a is the real semiaxis and c is the half of the interfocal distance, is called the eccentricity of a hyp erb ola. The eccentricity (11.1) is used in order to define one more parameter of a hyp erb ola. It is usually denoted through d: d= a2 a = . c (11.2)

Definition 11.2. On the plane of a hyp erb ola there are two lines p erp endicular to its real axis and placed at the distance d given by the formula (7.2) from its center. These lines are called directrices of a hyp erb ola.


180

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

Each hyp erb ola has two foci and two directrices. Each directrix has the corresp onding focus of it. This is that of two foci which is more close to the directrix in question. Let M (x, y ) b e some arbitrary p oint of a hyp erb ola. Let's connect this p oint with the left focus of the hyp erb ola F1 and drop the p erp endicular from it to the left directrix of the hyp erb ola. Let's denote through H1 the base of such a p erp endicular and calculate its length |M H1 |: |M H1 | = |x - (-d)| = |d + x|. (11.3)

Taking into account (11.2), the formula (11.3) can b e brought to |M H1 | = |a2 + c x| a2 +x = . c c (11.4)

The length of the segment M F1 was already calculated ab ove. Initially it was given by one of the formulas (10.10) , but later the more simple expression (10.25) was derived for it: |M F1 | = |a2 + c x| . a (11.5)

If we divide (11.5) by (11.4), we obtain the following relationship: c |M F1 | = = . |M H1 | a (11.6)

The p oint M can change its p osition on the hyp erb ola. Then the numerator and the denominator of the fraction (11.6) are


§ 12. . . . OF A TANGENT LINE TO A HYPERBOLA.

181

changed, but its value remains unchanged. This fact is known as the prop erty of directrices. Theorem 11.1. The ratio of the distance from some arbitrary p oint M of a hyp erb ola to its focus and the distance from this p oint to the corresp onding directrix is a constant which is equal to the eccentricity of the hyp erb ola. § 12. The equation of a tangent line to a hyp erb ola. Let's consider a hyp erb ola given by its canonical equation (10.20) in its canonical coordinate system (see Definition 10.4). Let's draw a tangent line to this hyp erb ola and denote through M = M (x0 , y0 ) its tangency p oint (see Fig. 12.1). Our goal is to write the equation of a tangent line to a hyp erb ola. A hyp erb ola consists of two branches, each branch b eing a curve comp osed by two halves -- the upp er half and the lower half. The upp er halves of the hyp erb ola branches can b e treated as a graph of some function of the form y = f (x) defined in the union of two intervals (-, -a) halves of a hyp erb ola can also b e treated as function of the form (12.1) with the same dom of a tangent line to the graph of the function the following well-known formula (see [9]): y = y0 + f (x0 ) (x - x0 ). (a, + a graph ain. The (12.1) is (12.1) ). Lower of some equation given by (12.2)

In order to apply the formula (12.2) to a hyp erb ola we need


182

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

to calculate the derivative of the function (12.1). Let's substitute the function (12.1) into the equation (10.20) : (f (x))2 x2 - = 1. a2 b2 (12.3)

The equality (12.3) is fulfilled identically in x. Let's differentiate the equality (12.3) with resp ect to x. This yields 2 x 2 f (x) f (x) - = 0. a2 b2 Let's apply (12.4) for to calculate the derivative f (x): f (x) = b2 x . a2 f (x) (12.5) (12.4)

In order to substitute (12.5) into the equation (12.2) we change x for x0 and f (x) for f (x0 ) = y0 . As a result we get f (x0 ) = b2 x0 . a2 y 0 (12.6)

Let's substitute (12.6) into the equation of the tangent line (12.2). This yields the following relationship y = y0 + b2 x 0 (x - x0 ). a2 y 0 (12.7)

Eliminating the denominator, we write the equality (12.7) as
2 a2 y y0 - b2 x x0 = a2 y0 - b2 x2 . 0

(12.8)

Now let's divide b oth sides of the equality (12.8) by a2 b2 :
2 x x0 y y0 x2 y0 0 - 2 = 2 - 2. a2 b a b

(12.9)

CopyRight c Sharipov R.A., 2010.


§ 12. . . . OF A TANGENT LINE TO A HYPERBOLA.

183

Note that the p oint M = M (x0 , y9 ) is on the hyp erb ola. Therefore its coordinates satisfy the equation (10.20):
2 y0 x2 0 - 2 = 1. a2 b

(12.10)

Taking into account (12.10), we can transform (12.9) to x x0 y y0 - 2 = 1. 2 a b (12.11)

Theorem 12.1. For a hyp erb ola determined by its canonical equation (10.20) its tangent line that touches this hyp erb ola at the p oint M = M (x0 , y0 ) is given by the equation (12.11). The equation (12.11) is a particular case of the equation (3.22) where the constants A, B , and D are given by the formulas A= x0 , a2 B=- y0 , b2 D = 1. (12.12)

According to the definition 3.6 and the formulas (3.21) the constants A and B in (12.12) are the covariant comp onents of the normal vector for the tangent line to a hyp erb ola. The tangent line equation (12.11) is written in a canonical coordinate system of a hyp erb ola. The basis of such a coordinate system is orthonormal. Therefore the formula (3.19) and the formula (32.4) from Chapter I yield the following relationships: A = n1 = n1 , B = n2 = n2 . (12.13)

Theorem 12.2. The quantities A and B in (12.12) are the coordinates of the normal vector n for the tangent line to a hyp erb ola which is given by the equation (12.11). The relationships (12.13) prove the theorem 12.2.


184

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

§ 13. Focal prop erty of a hyp erb ola. Like in the case of an ellipse, assume that a hyp erb ola is manufactured of a thin strip of some flexible material and assume that its surface is covered with a light reflecting layer. For such a hyp erb ola we can formulate the following focal prop erty. Theorem 13.1. A light ray emitted in one focus of a hyp erb ola up on reflecting on its surface goes to infinity so that its backward extension passes through the other focus of this hyp erb ola. Theorem 13.2. The tangent line of a hyp erb ola is a bisector in the triangle comp osed by the tangency p oint and two foci of the hyp erb ola. The theorem 13.2 is a purely geometric version of the theorem 13.1. These theorems are equivalent due to the reflection law saying that the angle of reflection is equal to the angle of incidence. Proof. Let's consider some p oint M = M (x0 , y0 ) on the hyp erb ola and draw the tangent line to the hyp erb ola through this p oint as shown in Fig. 13.1. Let N b e the x-intercept of this tangent line. Due to M and N , we have the segment [M N ]. This segment is p erp endicular to the normal vector n of the tangent line. In order to prove that [M N ] is a bisector in the triangle F1 M F2 it is sufficient to prove that n is directed along the bisector for the external angle of this triangle at its vertex M . This condition can b e written as -- - -- - (M F2 , n) (F1 M , n) = . |F1 M | |M F2 | (13.1)


§ 13. FOCAL PROPERTY OF A HYPERBOLA.

185

The coordinates of the p oints F1 and F2 in a canonical coordinate system of the hyp erb ola are known (see formulas (10.9)). The coordinates of the p oint M = M (x0 , y0 ) are also known. -- - Therefore we can find the coordinates of the vectors F1 M and -- - M F2 used in the ab ove formula (13.1): -- - x0 + c F1 M = , y0 -- - M F2 = c-x -y0
0

.

(13.2)

The tangent line that touches the hyp erb ola at the p oint M = = M (x0 , y0 ) is given by the equation (12.11). The coordinates of the normal vector n of this tangent line in Fig. 13.1 are given by the formulas (12.12) and (12.13) : x0 a2 (13.3) n= y0 -2 b Relying up on (13.2) and (13.3), we apply the formula (33.3) from Chapter I in order to calculate the scalar products in (13.1):
2 2 y0 c x0 x2 y0 c x0 + x2 -- - 0 0 - 2 = 2 + 2 - 2, (F1 M , n) = a2 b a a b 2 2 c x0 - x2 y0 c x0 x2 y0 -- - 0 0 (M F2 , n) = + 2 = 2 - 2 + 2. a2 b a a b

(13.4)

The coordinates of the p oint M satisfy the equation (10.20):
2 x2 y0 0 - 2 = 1. a2 b

(13.5)

Due to (13.5) the formulas (13.4) simplify to c x0 + a2 c x0 -- - , (F1 M , n) = 2 + 1 = a a2 c x0 c x0 - a2 -- - (M F2 , n) = 2 - 1 = . a a2

(13.6)


186

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

In order to calculate the denominators in the formula (13.1) we use the formulas (10.26) and (10.27) . In this case, when applied to the p oint M = M (x0 , y0 ), they yield |M F1 | = c |x0 | + sign(x0 ) a2 , a c |x0 | - sign(x0 ) a2 . |M F2 | = a

(13.7)

Due to the purely numeric identity |x0 | = sign(x0 ) x0 we can write (13.7) in the following form: |M F1 | = c x 0 + a2 sign(x0 ), a c x 0 - a2 sign(x0 ). |M F2 | = a

(13.8)

From the formulas (13.6) and (13.8) we easily derive the equalities -- - (F1 M , n) sign(x0 ) = , |M F1 | a -- - (M F2 , n) sign(x0 ) = |M F2 | a

that prove the equality (13.1) . The theorem 13.2 is proved. As we noted ab ove the theorem 13.1 is equivalent to the theorem 13.2 due to the light reflection law. Therefore the theorem 13.1 is also proved. § 14. Asymptotes of a hyp erb ola. Asymptotes are usually understood as some straight lines to which some p oints of a given curve come unlimitedly close along some unlimitedly long fragments of this curve. Each hyp erb ola has two asymptotes (see Fig. 14.1). In a canonical coordinate system the asymptotes of the hyp erb ola given by the equation


§ 15. PARABOLA.

187

(10.20) are determined by the following equations: b y = ± x. a (14.1)

One of the asymptotes is associated with the plus sign in the formula (14.1), the other asymptote is associated with the opp osite minus sign. The theory of asymptotes is closely related to the theory of limits. This theory is usually studied within the course of mathematical analysis (see [9]). For this reason I do not derive the equations (14.1) in this b ook. § 15. Parab ola. Canonical equation of a parab ola. Definition 15.1. A parabola is a set of p oints on some plane each of which is equally distant from some fixed p oint F of this plane and from some straight line d lying on this plane. The p oint F is called the focus of this parab ola, while the line d is called its directrix. Assume that a parab ola with the focus F and with the directrix d is given. Let's drop the p erp endicular from the p oint F onto the line d and denote through D the base of such a p erp endicular. Let's choose the


188

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

line D F for the x-axis of a coordinate system. Then we denote through O the midp oint of the segment [D F ] and choose this p oint O for the origin. And finally, we draw the y -axis of a coordinate system through the p oint O p erp endicular to the line D F (see Fig. 15.1). Choosing the unity scales along the axes, we ultimately fix a coordinate system with an orthonormal basis on the parab ola plane. Let's denote through p the distance from the focus F of the parab ola to its directrix d, i. e. we set |D F | = p. (15.1)

The p oint O is the midp oint of the segment [D F ]. Therefore the equality (15.1) leads to the following equalities: |D O| = |OF | = p . 2 (15.2)

The relationships (15.2) determine the coordinates of D and F : D = D (-p/2, 0), F = F (p/2, 0). (15.3)

Let M = M (x, y ) b e some arbitrary p oint of the parab ola. According to the definition 15.1, the following equality is fulfilled: |M F | = |M H | (15.4)

(see Fig. 15.1). Due to (15.3) the length of the segment [M F ] in the chosen coordinate system is given by the formula |M F | = y 2 + (x - p/2)2 ). (15.5)

The formula for the length of [M H ] is even more simple: |M H | = x + p/2. (15.6)


§ 15. PARABOLA.

189

Substituting (15.5) and (15.6) into (15.4), we get the equation y 2 + (x - p/2)2 ) = x + p/2. Let's square b oth sides of the equation (15.7): y 2 + (x - p/2)2 ) = (x + p/2)2 . (15.8) (15.7)

Up on expanding brackets and collecting similar terms in (15.8), we bring this equation to the following form: y 2 = 2 p x. (15.9)

Definition 15.2. The equality (15.9) is called the canonical equation of a parab ola. Theorem 15.1. For each p oint M (x, y ) of the parab ola determined by the initial equation (15.7) its coordinates satisfy the canonical equation (15.9). Due to (15.1) the constant p in the equation (15.9) is a nonnegative quantity. The case p = 0 corresp onds to the degenerate parab ola. From the definition 15.1 it is easy to derive that in this case the parab ola turns to the straight line coinciding with the x-axis in Fig. 15.1. The case of the degenerate parab ola is excluded by means of the inequality p > 0. (15.10)

Due to the inequality (15.10) from the equation (15.9) we derive x 0. (15.11)

Theorem 15.2. The canonical equation of the parab ola (15.9) is equivalent to the initial equation (15.7). Proof. In order to prove the theorem 15.2 it is sufficient
CopyRight c Sharipov R.A., 2010.


190

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

to invert the calculations p erformed in deriving the equality (15.9) from (15.7) . Note that the passage from (15.8) to (15.9) is invertible. The passage from (15.7) to (15.8) is also invertible due to the inequality (15.11), which follows from the equation (15.9). This observation completes the proof of the theorem 15.2. Definition 15.3. The p oint O in Fig. 15.1 is called the vertex of the parab ola, the line D F coinciding with the x-axis is called the axis of the parab ola. Definition 15.4. A coordinate system thonormal basis e1 , e2 where a parab ola is equation (15.9) and where the inequality called a canonical coordinate system of this O, e1 , e2 with an orgiven by its canonical (15.10) is fulfilled is parab ola.

§ 16. The eccentricity of a parab ola. The definition of a parab ola 15.1 is substantially different from the definition of an ellipse 6.1 and from the definition of a hyp erb ola 10.1. But it is similar to the prop erty of directrices of an ellipse in the theorem 7.1 and to the prop erty of directrices of a hyp erb ola in the theorem 11.1. Comparing the definition of a parab ola 15.1 with these theorems, we can formulate the following definition. Definition 16.1. The eccentricity of a parab ola is p ostulated to b e equal to the unity: = 1. § 17. The equation of a tangent line to a parab ola. Let's consider a parab ola given by its canonical equation (15.9) in its canonical coordinate system (see Definition 15.4). Let's draw a tangent line to this parab ola and denote through M = M (x0 , y0 ) the tangency p oint (see Fig. 17.1). Our goal is to write the equation of the tangent line to the parab ola through the p oint M = M (x0 , y0 ). An parab ola is a curve comp osed by two halves -- the upp er half and the lower half. Any one of these two halves of


§ 17. . . . TANGENT LINE TO A PARABOLA.

191

a parab ola can b e treated as a graph of some function y = f (x) (17.1)

with the domain (0, +). The equation of a tangent line to the graph of a function (17.1) is given by the well-known formula y - y0 = (17.2)

= f (x0 ) (x - x0 ).



(see [9]). In order to apply the formula (17.2) to a parab ola one should calculate the derivative of the function (17.1). Let's substitute (17.1) into the equation (15.9): (f (x))2 = 2 p x. (17.3) The equality (17.3) is fulfilled identically in x. Let's differentiate the equality (17.3) with resp ect to x. This yields 2 f (x) f (x) = 2 p. (17.4)

Let's apply the formula (17.4) for to calculate the derivative f (x) = p . f (x) (17.5)

In order to substitute (17.5) into the equation (17.2) we change x for x0 and f (x) for f (x0 ) = y0 . As a result we get f (x0 ) = p . y0 (17.6)

Let's substitute (17.6) into the equation of the tangent line


192

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

(17.2). This yields the following relationship: y - y0 = p (x - x0 ). y0 (17.7)

Eliminating the denominator, we write the equality (17.7) as
2 y y 0 - y 0 = p x - p x0 .

(17.8)

Note that the p oint M = M (x0 , y9 ) is on the parab ola. Therefore its coordinates satisfy the equality (15.9):
2 y 0 = 2 p x0 .

(17.9)

Taking into account (17.9), we can transform (17.8) to y y 0 = p x + p x0 . (17.10)

This is the required equation of a tangent line to a parab ola. Theorem 17.1. For a parab ola determined by its canonical equation (15.9) the tangent line that touches this parab ola at the p oint M = M (x0 , y0 ) is given by the equation (17.10). Let's write the equation of a tangent line to a parab ola in the following slightly transformed form: p x - y y0 + p x0 = 0. (17.11)

The equation (17.11) is a sp ecial instance of the equation (3.22) where the constants A, B , and D are given by the formulas A = p, B = -y0 , D = p x0 . (17.12)

According to the definition 3.6 and the formulas (3.21) , the constants A and B in (17.12) are the covariant comp onents of the normal vector of a tangent line to a parab ola. The


§ 18. FOCAL PROPERTY OF A PARABOLA.

193

equation (17.11) is written in a canonical coordinate system of the parab ola. The basis of a canonical system is orthonormal (see Definition 15.4). In the case of an orthonormal basis the formula (3.19) and the formula (32.4) from Chapter I yield A = n1 = n1 , B = n2 = n2 . (17.13)

Theorem 17.2. The quantities A and B in (17.12) are the coordinates of the normal vector n of a tangent line to a parab ola in the case where this tangent line is given by the equation (17.10). The relationships (17.13) prove the theorem 17.2. § 18. Focal prop erty of a parab ola. Assume that we have a parab ola manufactured of a thin strip of some flexible material covered with a light reflecting layer. For such a parab ola the following focal prop erty is formulated. Theorem 18.1. A light ray emitted from the focus of a parab ola up on reflecting on its surface goes to infinity parallel to the axis of this parab ola. Theorem 18.2. For any tangent line of a parab ola the triangle formed by the tangency p oint M , its focus F , and by the p oint N at which this tangent line intersects the axis of the parab ola is an isosceles triangle, i. e. the equality |M F | = |N F | holds. As we see in Fig. 18.1, the theorem 18.2 follows from the theorem 18.1 due to the reflection law saying that the angle of reflection is equal to the angle of


194

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

incidence and due to the equality of inner crosswise lying angles in the intersection of two parallel lines by a third line (see [6]). Proof of the theorem 18.2. For to prove this theorem we choose some tangency p oint M (x0 , y0 ) and write the equation of the tangent line to the parab ola in the form of (17.10) : y y 0 = p x + p x0 . (18.1)

Let's find the intersection p oint of the tangent line (18.1) with the axis of the parab ola. In a canonical coordinate system the axis of the parab ola coincides with the x-axis (see Definition 15.3). Substituting y = 0 into the equation (18.1), we get x = -x0 , which determines the coordinates of the p oint N : N = N (-x0 , 0). (18.2)

From (18.2) and (15.3) we derive the length of the segment [N F ]: |N F | = p/2 - (-x0 ) = p/2 + x0 . (18.3)

In the case of a parab ola the length of the segment [M F ] coincides with the length of the segment [M H ] (see Definition 15.1). Therefore from (15.3) we derive |M F | = |M H | = x0 - (-p/2) = x0 + p/2. (18.4)

Comparing (18.3) and (18.4) we get the required equality |M F | = |N F |. As a result the theorem 18.2 is proved. As for the theorem 18.1, it equivalent to the theorem 18.2. § 19. The scale of eccentricities. The eccentricity of an ellipse is determine by the formula (7.1), where the parameters c and a are related by the inequalities (6.4). Hence the eccentricity of an ellipse ob eys the inequalities 0 < 1. (19.1)


§ 20. CHANGING A COORDINATE SYSTEM.

195

The eccentricity of a parab ola is equal to unity by definition. Indeed, the definition 16.1 yields = 1. (19.2)

The eccentricity of a hyp erb ola is defined by the formula (11.1), where the parameters c and a ob ey the inequalities (10.8). Hence the eccentricity of a hyp erb ola ob eys the inequalities 1< Th e tricities from 0 scale of + . (19.3)

formulas (19.1) , (19.2) , and (19.3) show that the eccenof ellipses, parab olas, and hyp erb olas fill the interval to + without omissions, i. e. we have the continuous eccentricities. § 20. Changing a coordinate system.

~~ ~ ~ Let O, e1 , e2 , e3 and O, e1 , e2 , e3 b e two Cartesian coordinate systems in the space E. They consist of the bases e1 , e2 , e3 ~ ~~~ and e1 , e2 , e3 complemented with two p oints O and O, which are called origins (see Definition 1.1). The transition from the ~~~ basis e1 , e2 , e3 to the basis e1 , e2 , e3 and vice versa is describ ed by two transition matrices S and T whose comp onents are in the following transition formulas:
3 3 i S j ei , i =1

~ ej =

ej =
i =1

~ Tji e

i

(20.1)

(see formulas (22.4) and (22.9) in Chapter I). In order to describ e the transition from the coordinate system ~~ ~ ~ O, e1 , e2 , e3 to the coordinate system O, e1 , e2 , e3 and vice versa two auxiliary parameters are employed. These are the - - - - ~ ~ ~ vectors a = OO and a = OO .


196

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

- - - - ~ ~ ~ Definition 20.1. The vectors a = OO and a = OO are called the origin displacement vectors. - - ~ The origin displacement vector a = O O is usually expanded in the basis e1 , e2 , e3 , while the other origin displacement vector - - ~ ~ ~~~ a = OO is expanded in the basis e1 , e2 , e3 :
3 3 i

a=
i =1

a ei ,

~ a=
i =1

~~ ai e i .

(20.2)

- - - - ~ ~ ~ The vectors a = OO are a = OO opp osite to each other, i. e. the following relationships are fulfilled: ~ a = - a, ~ a = - a. (20.3)

Their coordinates in the expansions (20.2) are related with each other by means of the formulas
3 3

ai = - ~

Tji aj ,
j =1

ai = -

i Sj a j . ~ j =1

(20.4)

The formulas (20.4) are derived from the formulas (20.3) with the use of the formulas (25.4) and (25.5) from Chapter I. § 21. Transformation of the coordinates of a p oint under a change of a coordinate system. ~~ ~ ~ Let O, e1 , e2 , e3 and O, e1 , e2 , e3 b e two Cartesian coordinate systems in the space E and let X b e some arbitrary p oint in the space E. Let's denote through X = X (x1 , x2 , x3 ),
CopyRight c Sharipov R.A., 2010.

X = X (x1 , x2 , x3 ) ~~~

(21.1)


§ 22. ROTATION OF A COORDINATE SYSTEM . . .

197

the presentations of the p oint X in these two coordinate systems. The radius vectors of the p oint X in these systems are related with each other by means of the relationships r
X

= a + ~X , r

~X = a + rX , ~ r

(21.2)

The coordinates of X in (21.1) are the coordinates of its radius vectors in the bases of the corresp onding coordinate systems:
3 3

r

X

=
j =1

x ej ,

j

~X = r
j =1

x j ej . ~~

(21.3)

From (21.2), (21.3), and (20.2), applying the formulas (25.4) and (25.5) from Chapter I, we easily derive the following relationships:
3 3

xi = ~
j =1

Tji x j + ai , ~

xi =
j =1

i S j x j + ai . ~

(21.4)

Theorem 21.1. Under a change of coordinate systems in the space E determined by the formulas (20.1) and (20.2) the coordinates of p oints are transformed according to the formulas (21.4). The formulas (21.4) are called the direct and inverse transformation formulas for the coordinates of a p oint under a change of a Cartesian coordinate system. § 22. Rotation of a rectangular coordinate system on a plane. The rotation matrix. ~~ ~ Let O, e1 , e2 and O, e1 , e2 b e two Cartesian coordinate system on a plane. The formulas (20.1), (20.2), and (20.4) in this case are written as follows:
2 2 i S j ei , i =1

~ ej =

ej =
i =1

~ Tji ei ,

(22.1)


198

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.
2 2

a=
i =1

ai e i ,
2

~ a=
i =1

ai ei , ~~
2 i Sj a j . ~ j =1

(22.2) (22.3)

a =- ~

i

Tji
j =1

a,

j

a =-

i

Let X b e some arbitrary p oint of the plane. Its coordinates in ~~ ~ the coordinate systems O, e1 , e2 and O, e1 , e2 are transformed according to the formulas similar to (21.4):
2 2 i

x= ~
j =1

Tji

x +a , ~

j

i

x=
j =1

i

i Sj x j + ai . ~

(22.4)

~ Assume that the origins O and O do coincide. In this case 1 2 1 2 the parameters a , a and a , a in the formulas (22.2), (22.3), ~~ (22.3), and (22.4) do vanish. Un~ der the assumption O = O we consider the sp ecial case where ~~ the bases e1 , e2 and e1 , e2 b oth are orthonormal and where one of them is produced from the other by means of the rotation by some angle (see Fig. 22.1). For two bases on a plane the transition matrices S and T are square matrices 2 â 2. The comp onents of the direct transition matrix S are taken from the following formulas: ~ e1 = cos · e1 + sin · e2 , ~ e2 = - sin · e1 + cos · e2 . (22.5)

The formulas (22.5) are derived on the base of Fig. 22.1. Comparing the formulas (22.5) with the first relationship in 1 2 2 2 (22.1), we get S1 = cos , S1 = sin , S1 = - sin , S2 = cos .


§ 23. CURVES OF THE SECOND ORDER.

199

Hence we have the following formula for the matrix S : S= cos - sin sin cos . (22.6)

Definition 22.1. The square matrix of the form (22.6) is called the rotation matrix by the angle . The inverse transition matrix T is the inverse matrix for S (see theorem 23.1 in Chapter I). Due to this fact and due to the formula (22.6) we can calculate the matrix T : T= cos(-) - sin(-) sin(-) cos(-) . (22.7) But that that into

The matrix (22.7) is also a rotation matrix by the angle . the angle in it is taken with the minus sign, which means the rotation is p erformed in the opp osite direction. Let's write the relationships (22.4) taking into account ~ ~ a = 0 and a = 0, which follows from O = O, and taking account the formulas (22.6) and (22.7) : x1 = cos() x1 + sin() x2 , ~ x2 = - sin() x1 + cos() x2 , ~ x2 = sin() x1 + cos() x2 . ~ ~ x1 = cos() x1 - sin() x2 , ~ ~

(22.8)

(22.9)

The formulas (22.8) and (22.9) are the transformation formulas for the coordinates of a p oint under the rotation of the rectangular coordinate system shown in Fig. 22.1. § 23. Curves of the second order. Definition 23.1. A curve of the second order or a quadric


200

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

on a plane is a curve which is given by a p olynomial equation of the second order in some Cartesian coordinate system A x2 + 2 B x y + C y 2 + 2 D x + 2 E y + F = 0. (23.1)

Here x = x1 and y = x2 are the coordinates of a p oint on a plane. Note that the transformation of these coordinates under a change of one coordinate system for another is given by the functions of the first order in x1 and x2 (see formulas (22.4)). For this reason the general form of the equation of a quadric (23.1) does not change under a change of a coordinate system though the values of the parameters A, B , C , D , E , and F in it can change. From what was said we derive the following theorem. Theorem 23.1. For any curve of the second order on a plane, i. e. for any quadric, there is some rectangular coordinate system with an orthonormal basis such that this curve is given by an equation of the form (23.1) in this coordinate system. § 24. Classification of curves of the second order. Let b e a curve of the second order on a plane given by an equation of the form (23.1) in some rectangular coordinate system with the orthonormal basis. Passing from one of such coordinate systems to another, one can change the constant parameters A, B , C , D , E , and F in (23.1), and one can always choose a coordinate system in which the equation (23.1) takes its most simple form. Definition 24.1. The problem of finding a rectangular coordinate system with the orthonormal basis in which the equation of a curve of the second order takes its most simple form is called the problem of bringing the equation of a curve to its canonical form. An ellipse, a hyp erb ola, and parab ola are examples of curves of the second order on a plane. The canonical forms of the


§ 24. CLASSIFICATION OF CURVES . . .

201

equation (23.1) for these curves are already known to us (see formulas (6.15), (10.20) , and (15.9)). Definition 24.2. The problem of grouping curves by the form of their canonical equations is called the problem of classification of curves of the second order. Theorem 24.1. For any curve of the second order there is a rectangular coordinate system with an orthonormal basis where the constant parameter B of the equation (23.1) for is equal to zero: B = 0. Proof. Let O, e1 , e2 b e rectangular coordinate system with an orthonormal basis e1 , e2 where the equation of the curve has the form (23.1) (see Theorem 23.1). If B = 0 in (23.1), then O, e1 , e2 is a required coordinate system. If B = 0, then we p erform the rotation of the coordinate system O, e1 , e2 ab out the p oint O by some angle . Such a rotation is equivalent to the change of variables x = cos() x - sin() y , ~ ~ y = sin() x + cos() y ~ ~ (24.1)

in the equation (23.1) (see formulas (22.9) ). Up on substituting (24.1) into the equation (23.1) we get the analogous equation ~~ ~~~ ~~ ~~ ~~ ~ A x2 + 2 B x y + C y 2 + 2 D x + 2 E y + F = 0 (24.2)

~~~~~ ~ whose parameters A, B , C , D , E , and F are expressed through the parameters A, B , C , D , E , and F of the initial equation ~ (23.1). For the parameter B in (24.2) we derive the formula ~ B = (C - A) cos() sin() + B cos2 () - - B sin2 () = C-A sin(2 ) + B cos(2 ). 2

(24.3)


202

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

Since B = 0, we determine as a solution of the equation ctg(2 ) = A-C . 2B (24.4)

The equation (24.4) is always solvable in the form of = A-C 1 . - arctg 4 2 2B (24.5)

Comparing (24.4) and (24.3), we see that having determined the angle by means of the formula (24.5), we provide the vanishing ~ of the parameter B = 0 in the rotated coordinate system. The theorem 24.1 is proved. Let's apply the theorem 24.1 and write the equation of the curve in the form of the equation (23.1) with B = 0: A x2 + C y 2 + 2 D x + 2 E y + F = 0. (24.6)

The equation (24.6) provides the sub division of all curves of the second order on a plane into three typ es: ­ the elliptic typ e where A = 0, C = 0 and the parameters A and C are of the same sign, i. e. sign(A) = sign(C ); ­ the hyp erb olic typ e where A = 0, C = 0 and the parameters A and C are of different signs, i. e. sign(A) = sign(C ); ­ the parab olic typ e where A = 0 or C = 0. The parameters A and C in (24.6) cannot vanish simultaneously since in this case the degree of the p olynomial in (24.6) would b e lower than two, which would contradict the definition 23.1. Curves of the elliptic typ e. If the conditions A = 0, C = 0, and sign(A) = sign(C ) are fulfilled, without loss of generality we can assume that A > 0 and C > 0. In this case the equation (24.6) can b e written as follows: A x+ D A
2

+C y+

E C

2

+ F-

D2 E2 - A C

= 0.

(24.7)


§ 24. CLASSIFICATION OF CURVES . . .

203

Let's p erform the following change of variables in (24.7): x=x- ~ D , A y=y- ~ E . C (24.8)

The change of variables (24.8) corresp onds to the displacement of the origin without rotation (the case of the unit matrices S = 1 and T = 1 in (22.4)). In addition to (24.8) we denote E2 D2 ~ - . F =F- A C Taking into account (24.8) and (24.9), we write (24.7) as ~ A x2 + C y 2 + F = 0, ~ ~ (24.10) (24.9)

where the coefficients A and C are p ositive: A > 0 and C > 0. The equation (24.10) provides the sub division of curves of the elliptic typ e into three subtyp es: ~ ­ the case of an ellipse where F < 0; ~ ­ the case of an imaginary ellipse where F > 0; ~ = 0. ­ the case of a p oint where F In the case of an ellipse the equation (24.10) is brought to the ~ ~ form (6.15) in the variables x and y . As we know, this equation describ es an ellipse. In the case of an imaginary ellipse the equation (24.10) reduces to the equation which is similar to the equation of an ellipse: y2 ~ x2 ~ + 2 = -1. 2 a b (24.11)

The equation (24.11) has no solutions. Such an equation describ es the empty set of p oints. The case of a p oint is sometimes called the case of a pair of imaginary intersecting lines, which is somewhat not exact. In
CopyRight c Sharipov R.A., 2010.


204

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

this case the equation (24.10) describ es a single p oint with the coordinates x = 0 and y = 0. ~ ~ Curves of the hyp erb olic typ e. If the conditions A = 0, C = 0, and sign(A) = sign(C ) are fulfilled, without loss of generality we can assume that A > 0 and C < 0. In this case the equation (24.6) can b e written as A x+ D A
2

E ~ -C y- ~ C

2

+ F-

D2 E2 + ~ A C

= 0,

(24.12)

~ where -C = C > 0. Let's denote E2 D2 ~ + F =F- ~ A C (24.13)

and then p erform the following change of variables, which corresp onds to a displacement of the origin: x=x- ~ D , A y=y+ ~ E . ~ C (24.14)

Due to (24.13) and (24.14) the equation (24.12) is written as ~~ ~ A x2 - C y 2 + F = 0, ~ (24.15)

~ ~ where the coefficients A and C are p ositive: A > 0 and C > 0. The equation (24.15) provides the sub division of curves of the hyp erb olic typ e into two subtyp es: ~ ­ the case of a hyp erb ola where F = 0; ~ ­ the case of a pair of intersecting lines where F = 0. In the case of a hyp erb ola the equation (24.15) is brought ~ ~ to the form (10.20) in the variables x and y . As we know, it describ es a hyp erb ola. In the case of a pair of intersecting lines the left hand side of the equation (24.15) is written as a product of two multiplicands


§ 24. CLASSIFICATION OF CURVES . . .

205

and the equation (24.15) is brought to ~ ( Ax+ ~~ ~ C y )( A x - ~~ C y) = 0. (24.16)

The equation (24.16) describ es two line on a plane that intersect at the p oint with the coordinates x = 0 andy = 0. ~ ~ Curves of the parab olic typ e. For curves of this typ e there are two options in the equation (24.6) : A = 0, C = 0 or C = 0, A = 0. But the second option reduces to the first one up on changing variables x = -y, y = x, which corresp onds to ~ ~ the rotation by the angle = /2. Therefore without loss of generality we can assume that A = 0 and C = 0. Then the equation (24.6) is brought to y 2 + 2 D x + 2 E y + F = 0. (24.17)

In order to transform the equation (24.17) we apply the change of variables y = y - E , which corresp onds to the displacement of ~ ~ the origin along the y -axis. Then we denote F = F + E 2 . As a result the equation (24.17) is written as ~ y 2 + 2 D x + F = 0. ~ (24.18)

The equation (24.18) provides the sub division of curves of the parab olic typ e into four subtyp es: ­ the case ­ the case where D ­ the case where D ­ the case where D of of = of = of = a a 0 a 0 a 0 parab ola where D = 0; pair of parallel lines ~ and F < 0; pair of coinciding lines ~ and F = 0; pair of imaginary parallel lines ~ and F > 0.

In the case of a parab ola the equation (24.18) reduces to the equation (15.9) and describ es a parab ola.


206

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

In the case of a pair of parallel lines we introduce the notation 2 ~ F = -y0 into the equation (24.18). As a result the equation (24.18) is written in the following form: (y + y0 )(y - y0 ) = 0. ~ ~ (24.19)

The equation (24.19) describ es a pair of lines parallel to the y -axis and b eing at the distance 2 y0 from each other. In the case of a pair of coinciding lines the equation (24.18) reduces to the form y 2 = 0, which describ es a single line coinciding ~ with the y -axis. In the case of a pair of imaginary parallel lines the equation (24.18) has no solutions. It describ es the empty set. § 25. Surfaces of the second order. Definition 25.1. A surface of the second order or a quadric in the space E is a surface which in some Cartesian coordinate system is given by a p olynomial equation of the second order: A x2 + 2 B x y + C y 2 + 2 D x z + 2 E y z + + F z 2 + 2 G x + 2 H y + 2 I z + J = 0. (25.1)

Here x = x1 , y = x2 , z = x3 are the coordinates of p oints of the space E. Note that the transformation of the coordinates of p oints under a change of a coordinate system is given by functions of the first order in x1 , x2 , x3 (see formulas (21.4)). For this reason the general form of a quadric equation (25.1) remains unchanged under a change of a coordinate system, though the values of the parameters A, B , C , D , E , F , G, H , I , and J can change. The following theorem is immediate from what was said. Theorem 25.1. For any surface of the second order in the space E, i. e. for any quadric, there is some rectangular coordinate system with an orthonormal basis such that this surface is given


§ 26. CLASSIFICATION OF SURFACES . . .

207

by an equation of the form (25.1) in this coordinate system. § 26. Classification of surfaces of the second order. The problem of classification of surfaces of the second order in E is solved b elow following the scheme explained in § 2 of Chapter VI in the b ook [1]. Let S b e surface of the second order given by the equation (25.1) on some rectangular coordinate system with an orthonormal basis (see Theorem 25.1). Let's arrange the parameters A, B , C , D , E , F , G, H , I of the equation (25.1) into two matrices A F= B D B C E D E, F G D= H . I (26.1)

The matrices (26.1) are used in the following theorem. Theorem 26.1. For any surface of the second order S there is a rectangular coordinate system with an orthonormal basis such that the matrix F in (26.1) is diagonal, while the matrix D is related to the matrix F by means of the formula F · D = 0. The proof of the theorem 26.1 can b e found in [1]. Applying the theorem 26.1, we can write the equation (25.1) as A x2 + C y 2 + F z 2 + 2 G x + 2 H y + 2 I z + J = 0. (26.2)

The equation (26.2) and the theorem 26.1 provide the sub division of all surfaces of the second order in E into four typ es: ­ the elliptic typ e where A = 0, C = 0, F = 0 and the quantities A, C , and F are of the same sign; ­ the hyp erb olic typ e where A = 0, C = 0, F = 0 and the quantities A, C and F are of different signs; ­ the parab olic typ e where exactly one of the quantities A, C . and F is equal to zero and exactly one of the quantities


208

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

G, H , and I is nonzero. ­ the cylindrical typ e in all other cases. Surfaces of the elliptic typ e. From the conditions A = 0, C = 0, F = 0 in (26.2) and from the condition F · D = 0 in the theorem 26.1 we derive G = 0, H = 0, and I = 0. Since the quantities A, C , and F are of the same sign, without loss of generality we can assume that all of them are p ositive. Hence for all surfaces of the elliptic typ e we can write (26.2) as A x2 + C y 2 + F z 2 + J = 0, where A > 0, C > 0, and F > 0. The equation the sub division of surfaces of the elliptic typ e into ­ the case of an ellipsoid where J < 0; ­ the case of an imaginary ellipsoid where ­ the case of a p oint where J = 0. The case of an ellipsoid is the most non-trivial this case the equation (26.3) is brought to x2 y2 z2 + 2 + 2 = 1. a2 b c (26.3) (26.3) provides three subtyp es: J > 0; of the three. In

(26.4)

The equation (26.4) describ es the surface which is called an


§ 26. CLASSIFICATION OF SURFACES . . .

209

el lipsoid. This surface is shown in Fig. 26.1. In the case of an imaginary ellipsoid the equation (26.3) has no solutions. It describ es the empty set. In the case of a p oint the equation (26.3) can b e written in the form very similar to the equation of an ellipsoid (26.4): x2 y2 z2 + 2 + 2 = 0. a2 b c (26.5)

The equation (26.5) describ es a single p oint in the space E with the coordinates x = 0, y = 0, z = 0. Surfaces of the hyp erb olic typ e. From the three conditions A = 0, C = 0, F = 0 in (26.2) and from the condition F · D = 0 in the theorem 26.1 we derive G = 0, H = 0, and I = 0. The quantities A, C , and F are of different signs. Without loss of generality we can assume that two of them are p ositive and one of them is negative. By exchanging axes, which preserves the orthogonality of coordinate systems and orthonormality of their bases, we can transform the equation (26.2) so that we would have A > 0, C > 0, and F < 0. As a result we conclude that for all surfaces of the hyp erb olic typ e the initial equation (26.2) can b e brought to the form A x2 + C y 2 + F z 2 + J = 0, where A > 0, sub division of ­ the case ­ the case ­ the case (26.6)

C > 0, F < 0. The equation (26.6) provides the surfaces of the hyp erb olic typ e into three subtyp es: of a hyp erb oloid of one sheet where J < 0; of a hyp erb oloid of two sheets where J > 0; of a cone where J = 0.

In the case of a hyp erb oloid of one sheet the equation (26.6) can b e written in the following form: x2 y2 z2 + 2 - 2 = 1. a2 b c (26.7)


210

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

The equation (26.7) describ es a boloid of one sheet. It is shown In the case of a hyp erb oloid can b e written in the following

surface which is called the hyperin Fig. 26.2. of two sheets the equation (26.6) form: (26.8)

x2 y2 z2 + 2 - 2 = -1. a2 b c

The equation (26.8) describ es a surface which is called the hyperboloid of two sheets. This surface is shown in Fig. 26.3.

In the case of a cone the equation (26.6) is transformed to the equation which is very similar to the equations (26.7) and (26.8), but with zero in the right hand side: y2 z2 x2 + 2 - 2 = 0. a2 b c (26.9)

The equation (26.9) describ es a surface which is called the cone. This surface is shown in Fig. 26.4. Surfaces of the parab olic typ e. For this typ e of surfaces exactly one of the three quantities A, C , and F is equal to zero. By exchanging axes, which preserves the orthogonality of coordinate systems and orthonormality of their bases, we can
CopyRight c Sharipov R.A., 2010.


§ 26. CLASSIFICATION OF SURFACES . . .

211

transform the equation (26.2) so that we would have A = 0, C = 0, and F = 0. From A = 0, C = 0, and from the condition F · D = 0 in the theorem 26.1 we derive G = 0 and H = 0. The value of I is not determined by the condition F · D = 0. However, according to the definition of surfaces of the parab olic typ e exactly one of the three quantities G, H , I should b e nonzero. Due to G = 0 and H = 0 we conclude that I = 0. As a result the equation (26.2) is written as A x2 + C y 2 + 2 I z + J = 0, (26.10)

where A = 0, C = 0, and I = 0. The condition I = 0 means that we can p erform the displacement of the origin along the z -axis equivalent to the change of variables zz- J . 2I (26.11)

Up on applying the change of variables (26.11) to the equation (26.10) this equation is written as A x2 + C y 2 + 2 I z = 0, (26.12)

where A = 0, C = 0, I = 0. The equation (26.12) provides the sub division of surfaces of the parab olic typ e into two subtyp es: ­ the case of an elliptic parab oloid where the quantities A = 0 and C = 0 are of the same sign; ­ the case of a hyp erb olic parab oloid, where the quantities A = 0 and C = 0 are of different signs. In the case of an elliptic parab oloid the equation (26.12) can b e written in the following form: y2 x2 + 2 = 2 z. 2 a b (26.13)

The equation (26.13) describ es a surface which is called the


212

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

el liptic paraboloid. This surface is shown in Fig. 26.5. In the case of a hyp erb olic parab oloid the equation (26.12) can b e transformed to the following form: y2 x2 - 2 = 2 z. a2 b (26.14)

The equation (26.14) describ es a saddle surface which is called the hyperbolic paraboloid. This surface is shown in Fig. 26.6. Surfaces of the cylindrical typ e. According to the results from § 2 in Chapter VI of the b ook [1], in the cylindrical case the dimension reduction occurs. This means that there is a rectangular coordinate system with an orthonormal basis where the variable z drops from the equation (26.2): A x2 + C y 2 + 2 G x + 2 H y + J = 0. (26.15)

The classification of surfaces of the second order describ ed by the equation (26.15) is equivalent to the classification of curves of the second order on a plane describ ed by the equation (24.6). The complete typ e list of such surfaces contains nine cases: ­ the case of an elliptic cylinder;


§ 26. CLASSIFICATION OF SURFACES . . .

213

­ the case of an imaginary elliptic cylinder; ­ the case of a straight line;

­ ­ ­ ­ ­ ­

th th th th th th

e e e e e e

cas cas cas cas cas cas

e e e e e e

of of of of of of

a a a a a a

hyp erb olic cylinder; pair of intersecting planes; parab olic cylinder; pair of parallel planes; pair of coinciding planes; pair of imaginary parallel planes.

In the case of an elliptic cylinder the equation (26.15) can b e transformed to the following form: x2 y2 + 2 = 1. a2 b (26.16)

The equation (26.16) coincides with the equation of an ellipse on a plane (6.15) . In the space E it describ es a surface which is called the el liptic cylinder. This surface is shown in Fig. 26.7. In the case of an imaginary elliptic cylinder the equation


214

CHAPTER I I. GEOMETRY OF LINES AND SURFACES.

(26.15) can b e transformed to the following form: y2 x2 + 2 = -1. a2 b (26.17)

The equation (26.17) has no solutions. Such an equation describ es the empty set. In the case of a straight line the equation (26.15) can b e brought to the form similar to (26.16) and (26.17) : x2 y2 + 2 = 0. a2 b (26.18)

The equation (26.18) describ es a straight line in the space coinciding with the z -axis. In the canonical form this line is given by the equations x = 0 and y = 0 (see (5.14)). In the case of a hyp erb olic cylinder the equation (26.15) can b e transformed to the following form: x2 y2 - 2 = 1. a2 b (26.19)

The equation (26.19) coincides with the equation of a hyp erb ola on a plane (6.15) . In the spatial case it describ es a surface which is called the hyperbolic cylinder. It is shown in Fig. 26.8. The next case in the list is the case of a pair of intersecting planes. In this case the equation (26.15) can b e brought to x2 y2 - 2 = 0. a2 b (26.20)

The equation (26.20) describ es the union of two intersecting planes in the space given by the equations xy - = 0, a b xy + = 0. a b (26.21)


§ 26. CLASSIFICATION OF SURFACES . . .

215

The planes (26.21) intersect along a line which coincides with the z -axis. This line is given by the equations x = 0 and y = 0. In the case of a parab olic cylinder the equation (26.15) reduces to the equation coinciding with the equation of a parab ola y 2 = 2 p x. (26.22)

In the space the equation (26.22) describ es a surface which is called the parabolic cylinder. This surface is shown in Fig. 26.9. In the case of a pair of parallel planes the equation (26.15) is 2 brought to y 2 - y0 = 0, where y0 = 0. It describ es two parallel planes given by the equations y = y0 , y = -y0 . (26.23)

In the case of a pair of coinciding planes the equation (26.15) 2 is also brought to y 2 - y0 = 0, but the parameter y0 in it is equal to zero. Due to y0 = 0 two planes (26.23) are glued into a single plane which is p erp endicular to the y -axis and is given by the equation y = 0. In the case of a pair of imaginary parallel planes the equation 2 (26.15) is brought to y 2 + y0 = 0, where y0 = 0. Such an equation has no solutions. For this reason it describ es the empty set.


REFERENCES.

1. Sharipov R. A., Course of linear algebra and multidimensional geometry, Bashkir State University, Ufa, 1996; see also math.HO/0405323 in Electronic archive http:/ arXiv.org. / 2. Sharipov R. A., Course of differential geometry, Bashkir State University, Ufa, 1996; see also e-print math.HO/0412421 at http:/ arXiv.org. / 3. Sharipov R. A., Theory of representations of finite groups, Bash-NI I/ Stroy, Ufa, 1995; see also e-print math.HO/0612104 at http:/ arXiv.org. 4. Sharipov R. A., Classical electrodynamics and theory of relativity, Bashkir State University, Ufa, 1996; see also e-print physics/0311011 in Electronic archive http:/ arXiv.org. / 5. Sharipov R. A., Quick introduction to tensor analysis, free on-line textbo ok, 2004; see also e-print math.HO/0403252 at http:/ arXiv.org. / 6. Sharipov R. A., Foundations of geometry for university students and high-school students, Bashkir State University, 1998; see also e-print math.HO/0702029 in Electronic archive http:/ arXiv.org. / 7. Kurosh A. G., Course of higher algebra, Nauka publishers, Moscow, 1968. 8. Kronecker symbol, Wikipedia, the Free Encyclopedia, Wikimedia Foundation Inc., San Francisco, USA. 9. Kudryavtsev L. D., Course of mathematical analysis, Vol. I, I I, Visshaya Shkola publishers, Moscow, 1985.


CONTACTS

Address: Ruslan A. Sharip ov, Dep. of Mathematics and Information Techn., Bashkir State University, 32 Zaki Validi street, Ufa 450074, Russia Phone: +7-(347)-273-67-18 (Office) +7-(917)-476-93-48 (Cell) URL's: http:/ ruslan-sharip ov.ucoz.com / http:/ freetextb ooks.narod.ru / http:/ sovlit2.narod.ru /

Home address: Ruslan A. Sharip ov, 5 Rab ochaya street, Ufa 450003, Russia

E-mails: r-sharip ov@mail.ru R Sharip ov@ic.bashedu.ru

CopyRight c Sharipov R.A., 2010.


APPENDIX

List of publications by the author for the p eriod 1986­2011.
Part 1. Soliton theory. 1. Sharipov R. A., Finite-gap analogs of N -multiplet solutions of the KdV equation, Uspehi Mat. Nauk 41 (1986), no. 5, 203­204. 2. Sharipov R. A., Soliton multiplets of the Korteweg-de Vries equation, Dokladi AN SSSR 292 (1987), no. 6, 1356­1359. 3. Sharipov R. A., Multiplet solutions of the Kadomtsev-Petviashvili equation on a finite-gap background, Uspehi Mat. Nauk 42 (1987), no. 5, 221­222. 4. Bikbaev R. F., Sharipov R. A., Magnetization waves in Landau-Lifshits model, Physics Letters A 134 (1988), no. 2, 105-108; see solv-int/9905008. 5. Bikbaev R. F. & Sharipov R. A., Assymptotics as t for a solution of the Cauchy problem for the Korteweg-de Vries equation in the class of potentials with finite-gap behaviour as x ±, Theor. and Math. Phys. 78 (1989), no. 3, 345­356. 6. Sharipov R. A., On integration of the Bogoyavlensky chains, Mat. zametki 47 (1990), no. 1, 157­160. 7. Cherdantsev I. Yu. & Sharipov R. A., Finite-gap solutions of the Bullough-Dodd-Jiber-Shabat equation, Theor. and Math. Phys. 82 (1990), no . 1 , 1 5 5 ­ 1 6 0 . 8. Cherdantsev I. Yu. & Sharipov R. A., Solitons on a finite-gap background in Bul lough-Dodd-Jiber-Shabat model, International. Journ. of Mo dern Physics A 5 (1990), no. 5, 3021­3027; see math-ph/0112045. 9. Sharipov R. A. & Yamilov R. I., Backlund transformations and the construction of the integrable boundary value problem for the equation


LIST OF PUBLICATIONS.

219

10. 11. 12.

13.

14.

uxt = eu - e-2u , «Some problems of mathematical physics and asymptotics of its solutions», Institute of mathematics BNC UrO AN SSSR, Ufa, 1991, pp. 66­77; see solv-int/9412001. Sharipov R. A., Minimal tori in five-dimensional sphere in C3 , Theor. and Math. Phys. 87 (1991), no. 1, 48­56; see math.DG/0204253. Safin S. S. & Sharipov R. A., Backlund autotransformation for the equation uxt = eu - e-2u , Theor. and Math. Phys. 95 (1993), no. 1, 146­159. Boldin A. Yu. & Safin S. S. & Sharipov R. A., On an old paper of Tzitzeika and the inverse scattering method, Journal of Mathematical Physics 34 (1993), no. 12, 5801­5809. Pavlov M. V. & Svinolupov S. I. & Sharipov R. A., Invariant criterion of integrability for a system of equations of hydrodynamical type, «Integrability in dynamical systems», Inst. of Math. UrO RAN, Ufa, 1994, pp. 27­ 48; Funk. Anal. i Pril. 30 (1996), no. 1, 18­29; see solv-int/9407003. Ferapontov E. V. & Sharipov R. A., On conservation laws of the first order for a system of equations of hydrodynamical type, Theor. and Math. Phys. 108 (1996), no. 1, 109­128. Part 2. Geometry of the normal shift.

1. Boldin A. Yu. & Sharipov R. A., Dynamical systems accepting the normal shift, Theor. and Math. Phys. 97 (1993), no. 3, 386­395; see chaody n/ 9 4 0 3 0 0 3 . 2. Boldin A. Yu. & Sharipov R. A., Dynamical systems accepting the normal shift, Dokladi RAN 334 (1994), no. 2, 165­167. 3. Boldin A. Yu. & Sharipov R. A., Multidimensional dynamical systems accepting the normal shift, Theor. and Math. Phys. 100 (1994), no. 2, 264­269; see patt-sol/9404001. 4. Sharipov R. A., Problem of metrizability for the dynamical systems accepting the normal shift, Theor. and Math. Phys. 101 (1994), no. 1, 85­93; see solv-int/9404003. 5. Sharipov R. A., Dynamical systems accepting the normal shift, Uspehi Mat. Nauk 49 (1994), no. 4, 105; see solv-int/9404002. 6. Boldin A. Yu. & Dmitrieva V. V. & Safin S. S. & Sharipov R. A., Dynamical systems accepting the normal shift on an arbitrary Riemannian manifold, «Dynamical systems accepting the normal shift», Bashkir State University, Ufa, 1994, pp. 4­19; see also Theor. and Math. Phys. 103 (1995), no. 2, 256­266 and hep-th/9405021. 7. Boldin A. Yu. & Bronnikov A. A. & Dmitrieva V. V. & Sharipov R. A., Complete normality conditions for the dynamical systems on Riemannian manifolds, «Dynamical systems accepting the normal shift», Bashkir


220

LIST OF PUBLICATIONS. State University, 1994, pp. 20­30; see also Theor. and Math. Phys. 103 (1995), no. 2, 267­275 and astro-ph/9405049. Sharipov R. A., Higher dynamical systems accepting the normal shift, «Dynamical systems accepting the normal shift», Bashkir State University, 1994, pp. 41­65. Bronnikov A. A. & Sharipov R. A., Axial ly symmetric dynamical systems accepting the normal shift in Rn , «Integrability in dynamical systems», Inst. of Math. UrO RAN, Ufa, 1994, pp. 62­69. Sharipov R. A., Metrizability by means of a conformal ly equivalent metric for the dynamical systems, «Integrability in dynamical systems», Inst. of Math. UrO RAN, Ufa, 1994, pp. 80­90; see also Theor. and Math. Phys. 103 (1995), no. 2, 276­282. Boldin A. Yu. & Sharipov R. A., On the solution of the normality equations for the dimension n 3, Algebra i Analiz 10 (1998), no. 4, 31­61; see also solve-int/9610006. Sharipov R. A., Dynamical systems admitting the normal shift, Thesis for the degree of Do ctor of Sciences in Russia, math.DG/0002202, Electronic archive http:/ arXiv.org, 2000, pp. 1­219. / Sharipov R. A., Newtonian normal shift in multidimensional Riemannian geometry, Mat. Sbornik 192 (2001), no. 6, 105­144; see also math.DG /0006125. Sharipov R. A., Newtonian dynamical systems admitting the normal blow-up of points, Zap. semin. POMI 280 (2001), 278­298; see also math.DG/0008081. Sharipov R. A., On the solutions of the weak normality equations in multidimensional case, math.DG/0012110 in Electronic archive http:/ / arxiv.org (2000), 1­16. Sharipov R. A., First problem of globalization in the theory of dynamical systems admitting the normal shift of hypersurfaces, International Journal of Mathematics and Mathematical Sciences 30 (2002), no. 9, 541­557; see also math.DG/0101150. Sharipov R. A., Second problem of globalization in the theory of dynamical systems admitting the normal shift of hypersurfaces, math.DG /0102141 in Electronic archive http:/ arXiv.org (2001), 1­21. / Sharipov R. A., A note on Newtonian, Lagrangian, and Hamiltonian dynamical systems in Riemannian manifolds, math.DG/0107212 in Electronic archive http:/ arXiv.org (2001), 1­21. / Sharipov R. A., Dynamical systems admitting the normal shift and wave equations, Theor. and Math. Phys. 131 (2002), no. 2, 244­260; see also math.DG/0108158. Sharipov R. A., Normal shift in general Lagrangian dynamics, math.DG

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.


LIST OF PUBLICATIONS.

221

/0112089 in Electronic archive http:/ arXiv.org (2001), 1­27. / 21. Sharipov R. A, Comparative analysis for a pair of dynamical systems one / of which is Lagrangian, math.DG/0204161 in Electronic archive http:/ arxiv.org (2002), 1­40. 22. Sharipov R. A., On the concept of a normal shift in non-metric geometry, math.DG/0208029 in Electronic archive http:/ arXiv.org (2002), 1­47. / 23. Sharipov R. A., V -representation for the normality equations in geometry of generalized Legendre transformation, math.DG/0210216 in Electronic / archive http:/ arXiv.org (2002), 1­32. 24. Sharipov R. A., On a subset of the normality equations describing a generalized Legendre transformation, math.DG/0212059 in Electronic archive (2002), 1­19. Part 3. Mathematical analysis and theory of functions. 1. Sharipov R. A. & Sukhov A. B. On C R-mappings between algebraic Cauchy-Riemann manifolds and the separate algebraicity for holomorphic functions, Trans. of American Math. So ciety 348 (1996), no. 2, 767­780; see also Dokladi RAN 350 (1996), no. 4, 453­454. 2. Sharipov R. A. & Tsyganov E. N. On the separate algebraicity along families of algebraic curves, Preprint of Baskir State University, Ufa, 1996, pp. 1-7; see also Mat. Zametki 68 (2000), no. 2, 294­302. 3. Sharipov R. A., Algorithms for laying points optimal ly on a plane and a circle, e-print 0705.0350 in the archive http:/ arXiv.org (2010), 1­6. / 4. Sharipov R. A., A note on Khabibul lin's conjecture for integral inequali/ ties, e-print 1008.0376 in Electronic archive http:/ arXiv.org (2010), 1­17. 5. Sharipov R. A., Direct and inverse conversion formulas associated with Khabibul lin's conjecture for integral inequalities, e-print 1008.1572 in Electronic archive http:/ arXiv.org (2010), 1­7. / 6. Sharipov R. A., A counterexample to Khabibul lin's conjecture for integral inequalities, Ufa Math. Journ. 2 (2010), no. 4, 99­107; see also e-print 1008.2738 in Electronic archive http:/ arXiv.org (2010), 1­10. / Part 4. Symmetries and invariants. 1. Dmitrieva V. V. & Sharipov R. A., On the point transformations for the second order differential equations, solv-int/9703003 in Electronic archive http:/ arXiv.org (1997), 1­14. / 2. Sharipov R. A., On the point transformations for the equation y = P + 3 Q y + 3 R y 2 + S y 3 , solv-int/9706003 in Electronic archive http:/ / arxiv.org (1997), 1­35; see also V est n i k B ax k i r s k ogo un i v er s i t et a 1(I) (1998), 5­8.


222

LIST OF PUBLICATIONS.

3. Mikhailov O. N. & Sharipov R. A., On the point expansion for a certain class of differential equations of the second order, Diff. Uravneniya 36 (2000), no. 10, 1331­1335; see also solv-int/9712001. 4. Sharipov R. A., Effective procedure of point-classification for the equation y = P + 3 Q y + 3 R y 2 + S y 3 , math.DG/9802027 in Electronic archive http:/ arXiv.org (1998), 1­35. / 5. Dmitrieva V. V. & Gladkov A. V. & Sharipov R. A., On some equations that can be brought to the equations of diffusion type, Theor. and Math. Phys. 123 (2000), no. 1, 26­37; see also math.AP/9904080. 6. Dmitrieva V. V. & Neufeld E. G. & Sharipov R. A. & Tsaregoro dtsev A. A., On a point symmetry analysis for generalized diffusion type / equations, math.AP/9907130 in Electronic archive http:/ arXiv.org (1999), 1­52. Part 5. General algebra. 1. Sharipov R. A, Orthogonal matrices with rational components in composing tests for High School students, math.GM/0006230 in Electronic archive http:/ arXiv.org (2000), 1­10. / 2. Sharipov R. A., On the rational extension of Heisenberg algebra, math. RA/0009194 in Electronic archive http:/ arXiv.org (2000), 1­12. / 3. Sharipov R. A, An algorithm for generating orthogonal matrices with / rational elements, cs.MS/0201007 in Electronic archive http:/ arXiv.org (2002), 1­7. 4. Sharipov R. A, A note on pairs of metrics in a two-dimensional linear vector space, e-print 0710.3949 in Electronic archive http:/ arXiv.org / (2007), 1­9. 5. Sharipov R. A, A note on pairs of metrics in a three-dimensional linear vector space, e-print 0711.0555 in Electronic archive http:/ arXiv.org / (2007), 1­17. 6. Sharipov R. A., Transfinite normal and composition series of groups, e/ print 0908.2257 in Electronic archive http:/ arXiv.org (2010), 1­12; V est n i k B ax k i r s k ogo un i v er s i t et a 15 (2010), no. 4, 1098. 7. Sharipov R. A., Transfinite normal and composition series of modules, e-print 0909.2068 in Electronic archive http:/ arXiv.org (2010), 1­12. / Part 6. Condensed matter physics. 1. Lyuksyutov S. F. & Sharipov R. A., Note on kinematics, dynamics, and thermodynamics of plastic glassy media, cond-mat/0304190 in Electronic archive http:/ arXiv.org (2003), 1­19. /


LIST OF PUBLICATIONS.

223

2. Lyuksyutov S. F. & Paramonov P. B. & Sharipov R. A. & Sigalov G., Exact analytical solution for electrostatic field produced by biased atomic force microscope tip dwel ling above dielectric-conductor bilayer, condmat/0408247 in Electronic archive http:/ arXiv.org (2004), 1­6. / 3. Lyuksyutov S. F. & Sharipov R. A., Separation of plastic deformations in polymers based on elements of general nonlinear theory, cond-mat / /0408433 in Electronic archive http:/ arXiv.org (2004), 1­4. 4. Comer J. & Sharipov R. A., A note on the kinematics of dislocations in crystals, math-ph/0410006 in Electronic archive http:/ arXiv.org (2004), / 1­15. 5. Sharipov R. A., Gauge or not gauge? cond-mat/0410552 in Electronic / archive http:/ arXiv.org (2004), 1­12. 6. Sharipov R. A., Burgers space versus real space in the nonlinear theory of dislocations, cond-mat/0411148 in Electronic archive http:/ arXiv.org / (2004), 1­10. 7. Comer J. & Sharipov R. A., On the geometry of a dislocated medium, math-ph/0502007 in Electronic archive http:/ arXiv.org (2005), 1­17. / 8. Sharipov R. A., A note on the dynamics and thermodynamics of dislocated crystals, cond-mat/0504180 in Electronic archive http:/ arXiv.org / (2005), 1­18. 9. Lyuksyutov S. F., Paramonov P. B., Sharipov R. A., Sigalov G., Induced nanoscale deformations in polymers using atomic force microscopy, Phys. Rev. B 70 (2004), no. 174110. Part 7. Tensor analysis. 1. Sharipov R. A., Tensor functions of tensors and the concept of ex/ tended tensor fields, e-print math/0503332 in the archive http:/ arXiv.org (2005), 1­43. 2. Sharipov R. A., Spinor functions of spinors and the concept of extended spinor fields, e-print math.DG/0511350 in the archive http:/ arXiv.org / (2005), 1­56. 3. Sharipov R. A., Commutation relationships and curvature spin-tensors for extended spinor connections, e-print math.DG/0512396 in Electronic archive http:/ arXiv.org (2005), 1-22. / Part 8. Particles and fields. 1. Sharipov R. A., A note on Dirac spinors in a non-flat space-time of general relativity, e-print math.DG/0601262 in Electronic archive http: / arxiv.org (2006), 1­22. /


224

LIST OF PUBLICATIONS.

2. Sharipov R. A., A note on metric connections for chiral and Dirac spi/ nors, e-print math.DG/0602359 in Electronic archive http:/ arXiv.org (2006), 1­40. 3. Sharipov R. A., On the Dirac equation in a gravitation field and the secondary quantization, e-print math.DG/0603367 in Electronic archive http:/ arXiv.org (2006), 1­10. / 4. Sharipov R. A., The electro-weak and color bund les for the Standard Model in a gravitation field, e-print math.DG/0603611 in Electronic archive http:/ arXiv.org (2006), 1­8. / 5. Sharipov R. A., A note on connections of the Standard Model in a / gravitation field, e-print math.DG/0604145 in Electronic archive http:/ arxiv.org (2006), 1­11. 6. Sharipov R. A., A note on the Standard Model in a gravitation field, / e-print math.DG/0605709 in the archive http:/ arXiv.org (2006), 1­36. 7. Sharipov R. A., The Higgs field can be expressed through the lepton and quark fields, e-print hep-ph/0703001 in the archive http:/ arXiv.org / (2007), 1­4. 8. Sharipov R. A., Comparison of two formulas for metric connections in the bund le of Dirac spinors, e-print 0707.0482 in Electronic archive http:/ arXiv.org (2007), 1­16. / 9. Sharipov R. A., On the spinor structure of the homogeneous and isotropic / universe in closed model, e-print 0708.1171 in the archive http:/ arXiv.org (2007), 1­25. 10. Sharipov R. A., On Kil ling vector fields of a homogeneous and isotropic / universe in closed model, e-print 0708.2508 in the archive http:/ arXiv.org (2007), 1­19. 11. Sharipov R. A., On deformations of metrics and their associated spinor structures, e-print 0709.1460 in the archive http:/ arXiv.org (2007), 1­22. / 12. Sharipov R. A., A cubic identity for the Infeld-van der Waerden field and / its application, e-print 0801.0008 in Electronic archive http:/ arXiv.org (2008), 1­18. 13. Sharipov R. A., A note on Kosmann-Lie derivatives of Weyl spinors, / e-print 0801.0622 in Electronic archive http:/ arXiv.org (2008), 1­22. 14. Sharipov R. A., On operator fields in the bund le of Dirac spinors, e-print 0802.1491 in Electronic archive http:/ arXiv.org (2008), 1­14. / Part 9. Number theory. 1. Sharipov R. A., A note on a perfect Euler cuboid, e-print 1104.1716 in Electronic archive http:/ arXiv.org (2011), 1­8. / 2. Sharipov R. A., A note on the Sopfr(n) function, e-print 1104.5235 in Electronic archive http:/ arXiv.org (2011), 1­7. /

CopyRight c Sharipov R.A., 2010.


LIST OF PUBLICATIONS.

225

3. Sharipov R. A., Perfect cuboids and irreducible polynomials, e-print 1108. / 5348 in Electronic archive http:/ arXiv.org (2011), 1­8. 4. Sharipov R. A., A note on the first cuboid conjecture, e-print 1109.2534 in Electronic archive http:/ arXiv.org (2011), 1­6. / Part 10. Technologies and innovations. 1. Sharipov R. A., A polymer dental implant of the stocking type and its application, Russian patent RU 2401082 C2 (2009).


Uqebnoe izdanie XARIPOV Ruslan Abduloviq KURS ANALITIQESKO GEOMETRII

Uqebnoe posobie

Redaktor G. G. Sina ska Korrektor A. I. Nikolaeva

Licenzi na izdatelsku de telnost LR 021319 ot 05.01.1999 g. Podpisano v peqat 02.11.2010 g. Format 60â84/16. Usl. peq. l. 13,11. Uq.-izd . l. 11,62. Tira 100. Izd . 243. Zakaz 69 a. Redakcionno-izdatelski centr Baxkirskogo gosudarstvennogo universiteta 450074, RB, g. Ufa, ul. Zaki Validi, 32. Otpeqatano na mnoitelnom uqastke Baxkirskogo gosudarstvennogo universiteta 450074, RB, g. Ufa, ul. Zaki Validi, 32.