7.6 MULTIPLICATION OF VECTORS
O θφP
abcvFigure 7.11 The scalar triple product gives the volume of a parallelepiped.The scalar triple product is denoted by[a,b,c]≡a·(b×c)and, as its name suggests, it is just a number. It is most simply interpreted as the
volume of a parallelepiped whose edges are given bya,bandc(see figure 7.11).
The vectorv=a×bis perpendicular to the base of the solid and has magnitude
v=absinθ, i.e. the area of the base. Further,v·c=vccosφ. Thus, sinceccosφ
=OPis the vertical height of the parallelepiped, it is clear that (a×b)·c=area
of the base×perpendicular height = volume. It follows that, if the vectorsa,b
andcare coplanar,a·(b×c)=0.
Expressed in terms of the components of each vector with respect to theCartesian basis seti,j,kthe scalar triple product is
a·(b×c)=ax(bycz−bzcy)+ay(bzcx−bxcz)+az(bxcy−bycx),
(7.34)which can also be written as a determinant:
a·(b×c)=∣
∣
∣
∣
∣
∣ax ay az
bx by bz
cx cy cz∣
∣
∣
∣
∣
∣.By writing the vectors in component form, it can be shown thata·(b×c)=(a×b)·c,so that the dot and cross symbols can be interchanged without changing the result.
More generally, the scalar triple product is unchanged under cyclic permutation
of the vectorsa,b,c. Other permutations simply give the negative of the original
scalar triple product. These results can be summarised by
[a,b,c]=[b,c,a]=[c,a,b]=−[a,c,b]=−[b,a,c]=−[c,b,a]. (7.35)