2 properties of the dot product if a b and c are

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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 c 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: 1. a 3. a a b a2 1 c a2 2 a2 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 l line segments OA and OB in Figure 1. Note that if a and b are parallel vectors, then 0 or . The formula in the following theorem is used by physicists as the definition of the dot product. z B a-b b 0¨ 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 becomes 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: a b2 ab ab aa a ab 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 ab cos EXAMPLE 3 Find the...
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