Calculus: Analytic Geometry and Calculus, with Vectors

2.6 Vector products and changes of coordinates in E; 105

11 Prove that if u and v and w are vectors having scalar components ul,
u2, u3 and v1, 82, V3 and w1, w2, w3, then

X w) = (uli + u2j + u3k)

X w) =

u1

91

W1

U2 U3

V2 83

W2 W3

i j k

fl1^5393

W1 w2 w3

Remark: The number X w) is called the scalar triple product of the three
vectors, and we shall see how it is related to volumes of tetrahedrons and paral-
lelepipeds. Let v and w be nonzero nonparallel vectors having their tails at a
point .Q as in Figure 2.691. Then

(3) v Xw=lviJwJ sinOn=2IT2ln,

where 0 is the angle between v and w, n is the unit normal determined by the

right-hand rule, and I T21 is the area of the two dimensional triangle T2 of which

Figure 2.691

v and w form two sides. Let u be a third vector which has its tail at d and makes

the angle 0 with n. In case 0 < 0 < a/2, the number is the distance from

the tip of u to the plane of the vectors v and w. The volume V of the tetrahe-

dron having base T2 and opposite vertex B is therefore given by the formula

(4)

Hence

(5)

V =

V = x w).

The volume of the tetrahedron is half the volume of the pyramid whose vertex

is B and whose base is the parallelogram of which v and w are two adjacent sides.

The volume of this pyramid is one-third the volume V5 of the parallelepiped of

which u, v, w are adjacent edges. Therefore,

(6) Vp = X W).

Thus, when 0 < 4, < r/2, the scalar triple product X w) is the volume of

the parallelepiped of which u, v, w are three adjacent edges. When 0 = 7r/2,

the vector u lies in the plane of v and w and the scalar triple product is zero.