Mathematical Methods for Physics and Engineering : A Comprehensive Guide

VECTOR ALGEBRA

Find the volumeVof the parallelepiped with sidesa=i+2j+3k,b=4i+5j+6kand

c=7i+8j+10k.

We have already found thata×b=− 3 i+6j− 3 k, in subsection 7.6.2. Hence the volume
of the parallelepiped is given by

V=|a·(b×c)|=|(a×b)·c|

=|(− 3 i+6j− 3 k)·(7i+8j+10k)|

=|(−3)(7) + (6)(8) + (−3)(10)|=3.

Another useful formula involving both the scalar and vector products is La-

grange’s identity (see exercise 7.9), i.e.

(a×b)·(c×d)≡(a·c)(b·d)−(a·d)(b·c). (7.36)

7.6.4 Vector triple product

By the vector triple product of three vectorsa,b,cwe mean the vectora×(b×c).

Clearly,a×(b×c) is perpendicular toaand lies in the plane ofbandcand so

can be expressed in terms of them (see (7.37) below). We note, from (7.25), that

the vector triple product is not associative, i.e.a×(b×c)=(a×b)×c.

Two useful formulae involving the vector triple product are

a×(b×c)=(a·c)b−(a·b)c, (7.37)

(a×b)×c=(a·c)b−(b·c)a, (7.38)

which may be derived by writing each vector in component form (see exercise 7.8).

It can also be shown that for any three vectorsa,b,c,

a×(b×c)+b×(c×a)+c×(a×b)= 0.

7.7 Equations of lines, planes and spheres

Now that we have described the basic algebra of vectors, we can apply the results

to a variety of problems, the first of which is to find the equation of a line in

vector form.

7.7.1 Equation of a line

Consider the line passing through the fixed pointAwith position vectoraand

having a directionb(see figure 7.12). It is clear that the position vectorrof a

general pointRon the line can be written as

r=a+λb, (7.39)