5 theorem the vector a b is orthogonal to both a and

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Unformatted text preview: ND THE GEOMETRY OF SPACE One of the most important properties of the cross product is given by the following theorem. 5 Theorem The vector a b is orthogonal to both a and b. Proof In order to show that a b is orthogonal to a, we compute their dot product as follows: a b a2 b2 a a3 a1 b3 a1 b1 a3 a2 b3 a1 b1 a2 a3 b2 a 1 a 2 b3 a 3 b2 a 2 a 1 b3 a 3 b1 a 3 a 1 b2 a 1 a 2 b3 a 1 b2 a 3 a 1 a 2 b3 b1 a 2 a 3 a 2 b1 a 1 b2 a 3 b1 a 2 a 3 0 A similar computation shows that a both a and b. ¨ b 0. Therefore, a b is orthogonal to If a and b are represented by directed line segments with the same initial point (as in Figure 1), then Theorem 5 says that the cross product a b points in a direction perpendicular to the plane through a and b. It turns out that the direction of a b is given by the right-hand rule: If the ﬁngers of your right hand curl in the direction of a rotation (through an angle less than 180 ) from a to b, then your thumb points in the direction of a b. Now that we know the direction of the vector a b, the remaining thing we need to complete its geometric description is its length a b . This is given by the following theorem. axb a b b 6 Theorem If is the angle between a and b (so 0 ), then FIGURE 1 a Visual 13.4 shows how a as b changes. b changes b a Proof From the deﬁnitions of the cross product and length of a vector, we have a b 2 a 2 b3 a 3 b2 a2 b2 23 2 a 3 b1 a2 b2 32 2a 2 a 3 b2 b3 a2 b2 12 a2 1 a2 2 2a 1 a 2 b1 b2 a2 b2 3 1 a 2 b 2 ab a 2 b 2 a a 2 b 2 a 2 b2 2 b a 1 b3 2 a 1 b2 a2 b2 31 a 2 b1 2a 1 a 3 b1 b3 2 a2 b2 13 a2 b2 21 b2 3 a 1 b1 a 2 b2 a 3 b3 2 b 2 sin 2 1 2 2 b 2 cos 2 cos (by Theorem 13.3.3) 2 Taking square roots and observing that ssin 2 0 , we have ab a Geometric characterization of a b sin sin because sin 0 when b sin Since a vector is completely determined by its magnitude and direction, we can now say that a b is the vector that is perpendicular to both a and b, whose orientation is deter- 5E-13(pp 848-857) 1/18/06 11:20 AM Page 853 S ECTION 13.4 THE CROSS PRODUCT mined by the...
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This note was uploaded on 02/04/2010 for the course M 56435 taught by Professor Hamrick during the Fall '09 term at University of Texas at Austin.

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