Page 226 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

direction of rotation, then the thumb indicates (roughly) the direction of .

Figure 3.4.3
The reader may find it instructive to practice this rule with the products

Geometric Interpretation of Cross Product

If u and v are vectors in 3-space, then the norm of has a useful geometric interpretation. Lagrange's identity, given in
Theorem 3.4.1, states that

(5)

If θ denotes the angle between u and v, then , so 5 can be rewritten as

Since , it follows that , so this can be rewritten as

(6)

But is the altitude of the parallelogram determined by u and v (Figure 3.4.4). Thus, from 6, the area A of this
parallelogram is given by

Figure 3.4.4

This result is even correct if u and v are collinear, since the parallelogram determined by u and v has zero area and from 6 we have

because in this case. Thus we have the following theorem.

THEOREM 3.4.3