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Unformatted text preview: angle between the vectors a 2, 2, 1 and b 5, 3, 2 . 22 s38 SOLUTION Since a s2 2 22 1 2 3 and b s5 2 3 2 5E13(pp 838847) 1/18/06 11:16 AM Page 845 SECTION 13.3 THE DOT PRODUCT ❙❙❙❙ 845 and since
ab 25 2 3 12 2 we have, from Corollary 6,
ab
ab cos 2
3s38 So the angle between a and b is
cos 2
3 s38 1 1.46 or 84 Two nonzero vectors a and b are called perpendicular or orthogonal if the angle
between them is
2. Then Theorem 3 gives
ab a b cos 2 0 and conversely if a b 0, then cos
0, so
2. The zero vector 0 is considered
to be perpendicular to all vectors. Therefore, we have the following method for determining whether two vectors are orthogonal.
a and b are orthogonal if and only if a b 7 EXAMPLE 4 Show that 2 i 2j k is perpendicular to 5 i 4j 0.
2 k. SOLUTION Since 2i
a ¨ a b b b ¨ a · b >0 a · b =0 a · b <0 a 2j k 5i 4j 25 2 4 12 0 these vectors are perpendicular by (7).
Because cos
, we see that a b
0 if 0
2 and cos
0 if
2
is positive for
2 and negative for
2. We can think of a b as measuring
the extent to which a and b point in the same direction. The dot product a b is positive
if a and b point in the same general direction, 0 if they are perpendicular, and negative if
they point in generally opposite directions (see Figure 2). In the extreme case where a and
b point in exactly the same direction, we have
0, so cos
1 and FIGURE 2
Visual 13.3A shows an animation
of Figure 2. 2k ab a b If a and b point in exactly opposite directions, then
ab
a b. and so cos 1 and Direction Angles and Direction Cosines
The direction angles of a nonzero vector a are the angles , , and (in the interval 0,
that a makes with the positive x, y, and zaxes (see Figure 3 on page 846).
The cosines of these direction angles, cos , cos , and cos , are called the direction
cosines of the vector a. Using Corollary 6 with b replaced by i , we obtain
8 cos ai
ai a1
a 5E13(pp 838847) 846 ❙❙❙❙ 1/18/06 11:16 AM Page 846 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE z (This ca...
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 Fall '09
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