# Principles of vector calculus *Vectors. Opposite vectors. Zero vector. Length (modulus) of vector. * Collinear vectors. Coplanar vectors. Equality of vectors. Parallel transfer of vectors. Addition of vectors. Subtraction of vectors. Laws of addition of vectors. Laws of multiplication of vector by a number. Scalar product of vectors. Angle between non-zero vectors. Scalar square. Properties of a scalar product. Unit orthogonal vectors. Coordinates of a vector. Algebraic operations with vectors. Vector product of vectors. Properties of a vector product. Necessary and sufficient condition of collinearity of vectors. Necessary and sufficient condition of coplanarity of vectors. *Vector* is *a directed segment, connecting two points in a space ( in a plane ).* Vectors are signed usually eitherby small letters or by initial and final points. For example, a vector directed from point *A * to point *B* can be signed as * a*, * __* *A zero vector ***0*** *or 0* *is avector, for which initial and final points coincide, i.e. *A=B*.* *From here it follows: **0** =* – ***0**. * A length ( modulus ) of vector **a**, signed as | **a** |*,* *is a length of its imaging segment *AB*. Particularly, | **0** | = 0. Vectors are called *collinear** * ones, if their directed segments belong to parallel lines. Collinear vectors *a * and * b * are signed as * a * || * b* . Three or more vectors are called * coplanar*, if they lie in the same plane. * * *Equality of vectors. * Two vectors * a * and * b ** are equal*, if they are *collinear* ones and their *lengths are equal*, i.e. *a * || * b * and *| ** a** | = | **b** | .* Hence, *vectors don’t change ata parallel transfer.* *Addition of vectors. * As vectors are directed *segments*, then their addition can be executed *geometrically. *(Algebraic addition of vectors see below, in the point “Unit orthogonal vectors”). Assume, that __ __ *a** = AB* and * b* = *CD , * then __ __ * * *a *+ *b * = *AB *+ *CD* * * is a vector, received after executing of the two operations: *a*)* a parallel transfer* of one of thevectors till its initial point will coincide with a final point of another vector; *b*)* a geometrical addition *by drawing the resulting vector from an initial point of an immovable vector to a final point of a transferred vector. * * *Subtraction of vectors. * This operation is reduced to the previous by changing a subtracted vector to an oppositeone:
* a ** –* *b * =* a *+ ( *–* * b * ) . *Laws of addition. * *I. * * **a *+ *b * = * b* + *a * ( C o m m u t a t i v i t y ). *II.* ( *a * + *b * ) + * c* = *a * + ( * b* + *c * ) ( A s s o c i a t i v i t y ). *III.* *a * + **0*** * = * a .* * **IV.*** ** *a * + (*–* *a * )* = *** 0*** ** .* *Laws of multiplication of vector by a number.* * **I. *** **1 · *a * = * a * ,* *0 · *a * = 0 , *m * · **0** = **0** , ( *–*1 ) · *a * = *– ** a **.* * II. m **a** = **a** m *,* | m ** a * | = | *m* | · |* a | ** .* * III. m ( n **a **) = ( m n ) ** a ** . * ( A s s o c i a t i v i t y of * *multiplication by a number ). *IV. * ( *m + n* ) *a *= *m ** a** + n **a ** ,** *( D i s t r i b u t i v i t y of * **m * ( * a * + *b* )* * = *m ** a** + m **b ** . * multiplication by a number ). *__ __* *Scalar product of vectors. **Anangle between non-zero vectors* *AB* and *CD* is an angle, formed at a parallel transfer one of the vectors* * till coinciding the points *A* and *C. A scalarproduct of vectors **a * and *b *is called a number, equal to *a product of lengths ( modules )of these vectors by cosine of angle between them:* * * *If one of vectors is a zero vector, then a scalar product of these vectors is equal * *to zero by the definition:* * * (* a ,* **0*** * ) = (* *0* ,* *b *) = 0 . * * If both vectors are non-zero ones, then cosine of the angle between them may be found by the formula: A scalar product ( * a , a * ), equal to | * a* | ² , is called* a scalar square. * A length of vector * a* and its scalar square are tied by the relation: *A scalar product of two vectors is:* - *positive*, if an angle between the vectors is *acute *; - *negative,* if an angle between the vectors is *obtuse *. * * *A scalar product of two non-zero vectors is* *equal to zero*, *if and only if * *an angle between the vectors is right, i.e. these vectors are perpendicular* *( orthogonal ):* *Properties of a scalar product. *For any vectors *a ,* *b , c *and any number *m* the following relations are valid: * * * I.* ( *a ,* *b *) = (* b , a *) *.* ( Co m m u t a t i v i t y ) *II.* ( * m ***a ,** *b *) = * m * (* a ,* *b * ) *.* * III. * ( * a + b , c * ) = ( * a ,* *c *) + ( ** ***b ,* *c *)*. *( D i s t r i b u t i v i ty ) *Unit orthogonal vectors. *In anyrectangular system of coordinates it is possible to introduce *unit two-and-two orthogonal vectors **i*,* j *and* k*, connected with coordinate axes: * i * – for *x*-axis, * j * – for *y*-axis and * k* – for *z*-axis. According to this definition we have: ( *i* *, j * ) = ( * i* *, k * ) = ( * j* *, k * ) = 0, | *i | *=* | j | * =* | k | **=* 1. Any vector * a * can be expressed through these vectors by the only way: *a * = * x*** i + **y** j + **z** k ** .* *Another form of the record is: * a * = ( * x, y, z * )* *. * *Here* x*,* y*,* z – coordinates *of the vector* **a** *in this system of coordinates. According to the last relationand properties of the unit orthogonal vectors *i , j , k *a scalar product of two vectors can be written in another shape. Assume* a* = ( *x, y, z *); *b* = ( *u, v, w *). Then (* a ,* *b *) = * xu + yv + zw*. *A scalar product of two vectors is equal to a sum of products of corresponding * * coordinates. * A length (modulus) of vector ** ***a * = ( * x*,* **y*,* **z *) is equal to:
Besides, we receive now the possibility for *algebraic* operations with vectors; namely, addition and subtraction of vectors can be executed by coordinates: *a +* *b = *( *x + u , y + v , z + w *) ; *a *– *b = *( *x * –* u , y *– * v , z * –* w *) . *Vector product of vectors. **A vector product*** **[ *a ,* *b** *] *of vectors **a ** and ***b ** ( in the indicated order ) is a vector: There is another formula for a length of vector [ *a, b * ] : /\ | [ *a, b * ] | = | *a* | | *b* | sin ( * a, b** *)* *, i.e.* a length ( modulus ) of vector product of the vectors ** a ** and ** b ** is equal to * *product of lengths ( modules )of these vectors by sine of the angle between them.* Differently this fact can be interpreted as following : * a length ( modulus ) of vector* [ *a, b *] *is equal numerically to an area of parallelogram,built on vectors * *a ** and** b **.* * * *Properties of vector product.* *I. *A vector [ *a, b * ] is* perpendicular (orthogonal) both to vector* * a** *and *vector** b*. ( Prove this, please ! ) . *II. * [ *a ,* *b *] = *–* [* b , a *] . ( A n t i c o m m u t a t i v i t y ). *III.* [ * m ***a ,** *b *] = * m * [* a ,* *b *] .* * *IV.* [ *a + b , c *] = [ *a ,* *c *] + [ *b ,* *c *] .* *( D i s t r i b u t i v i t y ). *V. * [ *a ,* [ * b , c* ] ] = *b * ( * a , c * ) – * c * (* a , b *) . *VI.* [ [ * a ,* *b* ] *, c* ] = *b * ( * a , c * ) – * a * ( * b , c * ) . *Necessaryand sufficient condition of collinearity *of vectors *a* = ( *x, y, z *) and * b* = ( *u, v, w *) :* * Necessary and sufficient condition of coplanarity of vectors *a* = ( *x, y, z *) , *b* = ( *u, v, w *) and *c* = ( *p, q, r *) :* * * * E x a m p l e . The vectors: ** ***a* = ( 1, 2, 3 ) and *b* = ( – 2 , 0 ,4 ) are given. Calculate their scalar product, vector product and an angle between these vectors. S o l u t i o n . Using the corresponding formulas (see above), we’ll receive: a). the scalar product: (* a , b *) = 1 · ( – 2 ) + 2 · 0 + 3 · 4 = 10 ; b). the vector product: Back |