Page 208 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Figure 3.3.2

Component Form of the Dot Product

For purposes of computation, it is desirable to have a formula that expresses the dot product of two vectors in terms of the
components of the vectors. We will derive such a formula for vectors in 3-space; the derivation for vectors in 2-space is
similar.

Let and be two nonzero vectors. If, as shown in Figure 3.3.3, θ is the angle between u and v,
then the law of cosines yields

(2)

Since , we can rewrite 2 as

or

Substituting

and

we obtain, after simplifying,

(3)

Although we derived this formula under the assumption that u and v are nonzero, the formula is also valid if or
(verify).

Figure 3.3.3
If and are two vectors in 2-space, then the formula corresponding to 3 is