Page 222 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

There is an important difference between the dot product and cross product of two vectors—the dot product is a scalar and the
cross product is a vector. The following theorem gives some important relationships between the dot product and cross product
and also shows that is orthogonal to both u and v.
THEOREM 3.4.1

Relationships Involving Cross Product and Dot Product
If u, v, and w are vectors in 3-space, then

(a)

(b)

(c)

(d)

(e)

Proof (a) Let and . Then

Proof (b) Similar to (a).

Proof (c) Since
(2)

and
(3)

the proof can be completed by “multiplying out” the right sides of 2 and 3 and verifying their equality.