Mathematical Physics Vol 1

Chapter 1. Vector algebra

26

a × b

b

θ

a

b × a

Figure 1.13: Anticommutativity of a vector product.

It follows from the definition of a vector product that the vector product of two vectors of the same direction is equal to zero, i.e. a × α a = 0 . The previously given definition of a vector, with its corresponding operations, is a "geometric" definition. Namely, it follows from all the above that the vectors and the operations on them are independent of the choice of the coordinate system. In the text that follows, vectors will be observed “algebraically”, by defining their components with respect to a given coordinate system. The product of three vectors a · b × c = a · ( b × c ) , which is called mixed product is often used in practice. The product defined in this way is a scalar. It is obtained by initial vector multiplication of b and c , and then by scalar multiplication of the vector thus obtained and the vector a . The literature also uses the designation [ a , b , c ] for the product defined in this way. For a mixed product, the property of circular permutation applies, namely

[ a , b , c ]=[ b , c , a ]=[ c , a , b ] .

1.4.6 Reciprocal (conjugate) system of vectors

Definition Two sets of vectors a 1 ,..., a n and a ′

1 ,..., a ′ n are said to represent a reciprocal or conju gate system if the scalar product of a vector form one set with a vector from another is given by the relation a i · a ′ j = δ i j = 1 , i = j 0 , i̸ = j i , j = 1 ,..., n , (1.25)

which can also be represented by the following table (for n = 3) of by the following figure, for n = 2 (Fig. 1.14).