Mathematical Physics Vol 1

Chapter 1. Vector algebra

32

which follows from the assumption that the base vectors are orthonormal.

Vector product

i j k a x a y a z b x b y b z

a × b =

(1.46)

.

The vector product can be represented by the above symbolic determinant, as shown in exercise 4, on p. 56. Namely, the vector product a × b represented in this way is equal to the expression obtained by developing this determinant by the first row.

Mixed product

a x a y a z b x b y b z c x c y c z

a · ( b × c )=

(1.47)

.

Note that a mixed product is zero if these three vectors are coplanar, i.e. linearly dependent. In particular, if a is collinear, with say, b then a = λ b , and by substitution in (1.47) we obtain

a x a z λ a x λ a y λ a z c x c y c z a y

a x a y a z a x a y a z c x c y c z

a · ( λ a × c )=

= λ

= 0 .

We have used here the following properties of a determinant - a determinant is multiplied by a number by multiplying elements of one row or column by that number and - a determinant is equal to zero if any two rows or columns are equal. We would obtain the result in a similar way in the case of collinearity of vectors a and c . 1.5 Algebraic model of linear vector space We are familiar with the use of "ordinary" vectors in three-dimensional space to represent physical quantities, such as: position vector, velocity, acceleration, force, etc. We will now define an abstract linear vector space using the well-known properties of such vectors. Definition Let X be a nonempty set whose elements x , y , z ,..., will be called vectors, and T a set of all real (complex) numbers, whose elements α , β ,..., will be called scalars. The pair ( X , T ) forms a linear vector space or shortly vector space (real or complex, depending on the set of scalars T), if it has the following algebraic structure

i) to each ordered pair of vectors ( x , y ) from X corresponds a third vector X , which will be called their sum, and denoted by x + y . The operation that assigns vector x + y to the ordered pair ( x , y ) will be called addition of vectors . ii) To each vector x ∈ X and each scalar α ∈ T corresponds a vector from X , which will be called product of the vector x by the scalar α , and denoted by α x . The operation that associates vector x from X and scalar α to the vector α x is called multiplication of a vector by a scalar .

The operations addition of vectors and multiplication of a vector by a scalar have the following properties