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Unformatted text preview: 2 The dot product obeys many of the laws that hold for ordinary products of real numbers. These are stated in the following theorem.
2 Properties of the Dot Product If a, b, and c are vectors in V3 and c is a scalar, then 1. a a
3. a b
5. 0 a a
0 2 2. a ab ac b 4. c a b ba
ca b a cb These properties are easily proved using Deﬁnition 1. For instance, here are the proofs
of Properties 1 and 3:
3. a a
3 a a1, a2, a3 2 b1 c1, b2 c2, b3 c3 a 3 b3 c3 a 1 b1 c1 a 2 b2 c2 a 1 b1 a 1 c1 a 2 b2 a 2 c2 a 1 b1 a 2 b2 a 3 b3 a 1 c1 a 3 b3
a 2 c2 a 3 c3
a 3 c3 ab ac
The proofs of the remaining properties are left as exercises.
The dot product a b can be given a geometric interpretation in terms of the angle
between a and b, which is deﬁned to be the angle between the representations of a and
. In other words, is the angle between the
b that start at the origin, where 0 5E-13(pp 838-847) 844 ❙❙❙❙ 1/18/06 11:15 AM Page 844 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE l
line segments OA and OB in Figure 1. Note that if a and b are parallel vectors, then
The formula in the following theorem is used by physicists as the deﬁnition of the dot
product. z B
x a A
3 Theorem If is the angle between the vectors a and b, then
ab y a b cos FIGURE 1 Proof If we apply the Law of Cosines to triangle OAB in Figure 1, we get AB 4 2 OA 2 2 OB 2 OA OB cos (Observe that the Law of Cosines still applies in the limiting cases when
0 or , or
a 0 or b 0.) But OA
a , OB
b , and AB
a b , so Equation 4
a 5 2 b a 2 b 2 2a b cos Using Properties 1, 2, and 3 of the dot product, we can rewrite the left side of this equation as follows:
2a b 2 ba
b bb 2 Therefore, Equation 5 gives
a 2 2a b Thus b 2 a 2a b a 2 b 2a ab or 2 2a b cos b cos b cos EXAMPLE 2 If the vectors a and b have lengths 4 and 6, and the angle between them is 3, ﬁnd a b.
SOLUTION Using Theorem 3, we have ab a b cos 3 46 1
2 12 The formula in Theorem 3 also enables us to ﬁnd the angle between two vectors.
6 Corollary If is the angle between the nonzero vectors a and b, then
ab cos EXAMPLE 3 Find the...
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- Fall '09