James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

810 Chapter 12 Vectors and the Geometry of Space
a
¨
a
¨
a
FIGURE 2
b
b
b
a · b>0
¨ acute
a · b=0
¨=π/2
a · b<0
¨ obtuse
Because cos . 0 if 0 < , y2 and cos , 0 if y2 , < , we see that
a ? b is positive for , y2 and negative for . y2. 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
a ? b − | a | | b |
If a and b point in exactly opposite directions, then we have − and so cos − 21
and a ? b − 2| a | | b | .
TEC Visual 12.3A shows an animation
of Figure 2.
x
a¡
z
FIGURE 3
ç
å
a
∫
y
Direction Angles and Direction Cosines
The direction angles of a nonzero vector a are the angles , , and (in the interval
f0, gd that a makes with the positive x-, y-, and z-axes, respectively. (See Figure 3.)
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 − a ? i
| a | | i | − a 1
| a |
(This can also be seen directly from Figure 3.)
Similarly, we also have
9 cos − a 2
| a |
cos − a 3
| a |
By squaring the expressions in Equations 8 and 9 and adding, we see that
10 cos 2 1 cos 2 1 cos 2 − 1
We can also use Equations 8 and 9 to write
Therefore
11
a − k a 1 , a 2 , a 3 l − k| a | cos , | a | cos , | a | cos l
− | a | kcos , cos , cos l
1
| a a − k cos , cos , cos l
|
which says that the direction cosines of a are the components of the unit vector in the
direction of a.
ExamplE 5 Find the direction angles of the vector a − k 1, 2, 3 l.
SOLUtion Since | a | − s1 2 1 2 2 1 3 2 − s14 , Equations 8 and 9 give
and so
cos − 1
s14
cos − 2
s14
cos − 3
s14
− cos 21S 1
s14
D < 74° − cos 21S 2
s14
D < 58° − cos 21S 3
s14
D < 37°
■
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.