Page 225 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Figure 3.4.1
The standard unit vectors.

The reader should have no trouble obtaining the following results:

Figure 3.4.2 is helpful for remembering these results. Referring to this diagram, the cross product of two consecutive vectors going
clockwise is the next vector around, and the cross product of two consecutive vectors going counterclockwise is the negative of the
next vector around.

Figure 3.4.2

Determinant Form of Cross Product

It is also worth noting that a cross product can be represented symbolically in the form of a formal 3 × 3 determinant:

(4)

For example, if and , then

which agrees with the result obtained in Example 1. . For example,
Warning It is not true in general that

and

so

We know from Theorem 3.4.1 that is orthogonal to both u and v. If u and v are nonzero vectors, it can be shown that the
direction of can be determined using the following “right-hand rule”* (Figure 3.4.3): Let θ be the angle between u and v, and
suppose u is rotated through the angle θ until it coincides with v. If the fingers of the right hand are cupped so that they point in the