Page 211 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Therefore, u and v make an obtuse angle, v and w make an acute angle, and u and w are perpendicular.

Orthogonal Vectors

Perpendicular vectors are also called orthogonal vectors. In light of Theorem 3.3.1 b, two nonzero vectors are orthogonal if

and only if their dot product is zero. If we agree to consider u and v to be perpendicular when either or both of these vectors is

, then we can state without exception that two vectors u and v are orthogonal (perpendicular) if and only if . To

indicate that u and v are orthogonal vectors, we write .

EXAMPLE 5 A Vector Perpendicular to a Line

Show that in 2-space the nonzero vector is perpendicular to the line .

Solution be distinct points on the line, so that

Let and

Since the vector runs along the line (Figure 3.3.5), we need only show that n and (6)
are

perpendicular. But on subtracting the equations in (6), we obtain

which can be expressed in the form

Thus n and are perpendicular.

Figure 3.3.5
The following theorem lists the most important properties of the dot product. They are useful in calculations involving vectors.
THEOREM 3.3.2

Properties of the Dot Product