Page 213 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

The following theorem gives formulas for calculating and .
THEOREM 3.3.3

If u and a are vectors in 2-space or 3-space and if , then

Proof Let and . Since is parallel to a, it must be a scalar multiple of a, so it can be written in
the form . Thus

(8)

Taking the dot product of both sides of 8 with a and using Theorems Theorem 3.3.1a and Theorem
3.3.2 yields

(9)

But since is perpendicular to a; so 9 yields

Since , we obtain

EXAMPLE 6 Vector Component of u Along a
Let and . Find the vector component of u along a and the vector component of u orthogonal to
a.

Solution

Thus the vector component of u along a is
and the vector component of u orthogonal to a is