Mathematical Methods for Physics and Engineering - Matematica.NET

7.6 MULTIPLICATION OF VECTORSvPOφθcbaFigure 7.11The 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 thevolume of a parallelepiped whose edges are given by a, b and c (see figure 7.11).The vector v = a × b is perpendicular to the base of the solid and has magnitudev = ab sin θ, i.e. the area of the base. Further, v · c = vc cos φ. Thus, since c cos φ= OP is the vertical height of the parallelepiped, it is clear that (a × b) · c =areaof the base × perpendicular height = volume. It follows that, if the vectors a, band c are coplanar, a · (b × c) =0.Expressed in terms of the components of each vector with respect to theCartesian basis set i, j, k the scalar triple product isa · (b × c) =a x (b y c z − b z c y )+a y (b z c x − b x c z )+a z (b x c y − b y c x ),(7.34)which can also be written as a determinant:∣ a x a y a z ∣∣∣∣∣a · (b × c) =b x b y b z .∣ c x c y c zBy 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 permutationof the vectors a, b, c. Other permutations simply give the negative of the originalscalar 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)225