Mathematics of Physics and Engineering

(^22) Vector Operations
product formulas (1.2.3) and (1.2.6) are usually more convenient for angle
computations.
Remark 1.4 From (1.2.16) it follows that (Aw) x v = X(u x v) = u x
At; for any scalar A. Another consequence of (1.2.16) is the distributive
property of the cross product:
rx(u + v) — rxu + rxv. (1.2.25)
Still, the cross product is not associative; instead, the following identity
holds:
u x (v x w) + v x (w x u) + w x (u x v) = 0.
EXERCISE 1.2.11."^4 Prove that
u x (v x w) = (u • w)v — (u • v)w.
(1.2.26)
(1.2.27)
Then use the result to verify (1.2.26). Hint: A possible proof of (1.2.27) is
as follows (fill in the details). Choose an orthonormal basis t, j, k so that i is
parallel to w and j is in the plane of w and v. Then w = w\l and v = v%i + v-z]
and
v x w = det = —V2W1K,;
) = det
i 3 it
V\ V2 0
101 0 0
1 j k
U\ «2 U3
0 0 —V2W1
= —U2V2W11 + U1V2W1J;
(u -w)v — (u -v)lV = UlWl(v\l + V2J) — (uiVl +U2«2)Wl* = — U2V2W1I + U\V2W\j.
While the properties (Cl)-(C4) of the cross product are independent of
the coordinate system, the definition does not generalize to M.n for n > 4
because in dimension n > 4 there are too many vectors orthogonal to two
given vectors.
Property (CI) implies that ||u x v|| is the area of the parallelogram
generated by the vectors u and v. Accordingly, we have u x v = 0 if and
only if one of the vectors is a scalar multiple of the other. If Pi, P2, P3
are three points in E^3 , these points are collinear (lie on the same line) if
and only if
P 1 P 2 x PXP 3 = 0, (1.2.28)