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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
righthand 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 5E13(pp 848857) 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.
 Fall '09
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