Unformatted text preview: n also be seen directly from Figure 3.) Similarly, we also have ç
a¡ a
∫ a2
a cos 9 a3
a cos By squaring the expressions in Equations 8 and 9 and adding, we see that å y cos 2 10 cos 2 cos 2 1 x We can also use Equations 8 and 9 to write FIGURE 3 a a 1, a 2 , a 3 a cos , a cos , a cos a cos , cos , cos
Therefore
1
a
a 11 cos , cos , cos 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
SOLUTION Since a s1 2 2 2 3 1
s14 cos 2 1, 2, 3 . s14, Equations 8 and 9 give cos 2
s14 3
s14 cos and so
cos 1 1
s14 74 cos 1 2
s14 58 cos 1 3
s14 37 Projections
l
l
Figure 4 shows representations PQ and PR of two vectors a and b with the same initial
l
point P. If S is the foot of the perpendicular from R to the line containing PQ, then the
l
vector with representation PS is called the vector projection of b onto a and is denoted
by proja b.
R
Visual 13.3B shows how Figure 4
changes when we vary a and b. R
b b a a
FIGURE 4 Vector projections P S proj a b Q
S P Q proj a b The scalar projection of b onto a (also called the component of b along a) is deﬁned
to be the magnitude of the vector projection, which is the number b cos , where is the 5E13(pp 838847) 1/18/06 11:16 AM Page 847 SECTION 13.3 THE DOT PRODUCT b P 847 angle between a and b. (See Figure 5; you can think of the scalar projection of b as being
the length of a shadow of b.) This is denoted by compa b. Observe that it is negative if
2
. The equation R a ¨ ❙❙❙❙ ab Q S b cos ¨ a b cos a ( b cos ) shows that the dot product of a and b can be interpreted as the length of a times the scalar
projection of b onto a. Since FIGURE 5 Scalar projection ab
a b cos a
a b the component of b along a can be computed by taking the dot product of b with the unit
vector in the direction of a. We summarize these ideas as follows. Scalar projection of b onto a: compa b ab
a Vector projection of b onto a: proja b ab
a a
a ab
a
a2 Notice that the vector projection is the scalar projection times the unit vector in the direction of a.
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 Fall '09
 hamrick
 Linear Algebra, Vectors, Vector Space, Dot Product, (pp 828837)

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