Mathematical Methods for Physics and Engineering - Matematica.NET

VECTOR ALGEBRA◮Find the volume V of the parallelepiped with sides a = i +2j +3k, b =4i +5j +6k andc =7i +8j +10k.We have already found that a × b = −3i +6j − 3k, in subsection 7.6.2. Hence the volumeof the parallelepiped is given byV = |a · (b × c)| = |(a × b) · c|= |(−3i +6j − 3k) · (7i +8j +10k)|= |(−3)(7) + (6)(8) + (−3)(10)| =3. ◭Another useful formula involving both the scalar and vector products is Lagrange’sidentity (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 productBy the vector triple product of three vectors a, b, c we mean the vector a × (b × c).Clearly, a × (b × c) is perpendicular to a and lies in the plane of b and c and socan be expressed in terms of them (see (7.37) below). We note, from (7.25), thatthe vector triple product is not associative, i.e. a × (b × c) ≠(a × b) × c.Two useful formulae involving the vector triple product area × (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 vectors a, b, c,a × (b × c)+b × (c × a)+c × (a × b) =0.7.7 Equations of lines, planes and spheresNow that we have described the basic algebra of vectors, we can apply the resultsto a variety of problems, the first of which is to find the equation of a line invector form.7.7.1 Equation of a lineConsider the line passing through the fixed point A with position vector a andhaving a direction b (see figure 7.12). It is clear that the position vector r of ageneral point R on the line can be written asr = a + λb, (7.39)226