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Unformatted text preview: i Thus, the cross product is not commutative. Also
i i j i k 0 j j whereas
i i j 0 So the associative law for multiplication does not usually hold; that is, in general,
a b c a b c However, some of the usual laws of algebra do hold for cross products. The following theorem summarizes the properties of vector products.
8 Theorem If a, b, and c are vectors and c is a scalar, then
1. a 2.
3.
4.
5.
6. b
b b
(c a)
c (a
a ( b c) a
(a b) c a
abc
a
a
bc
a a
b)
b
c
b
cb a
a
b
c (c b)
c
c
a bc These properties can be proved by writing the vectors in terms of their components
and using the deﬁnition of a cross product. We give the proof of Property 5 and leave the
remaining proofs as exercises.
Proof of Property 5 If a b c b1, b2 , b3 , and c a 1 b2 c3 b3 c2 a 2 b3 c1 b1 c3 a 1 b2 c3 a 1 b3 c2 a 2 b3 c1 a 2 b1 c3 a 2 b3 9 a a 1, a 2 , a 3 , b a 3 b2 c1 a 3 b1 a b a 1 b3 c2 c1, c2 , c3 , then
a 3 b1 c2
a 3 b1 c2
a 1 b2 b2 c1
a 3 b2 c1
a 2 b1 c3 c The product a b c that occurs in Property 5 is called the scalar triple product of
the vectors a, b, and c. Notice from Equation 9 that we can write the scalar triple product
as a determinant:
10 a b c a1
b1
c1 a2
b2
c2 a3
b3
c3 5E13(pp 848857) 1/18/06 11:21 AM Page 855 SECTION 13.4 THE CROSS PRODUCT ❙❙❙❙ 855 The geometric signiﬁcance of the scalar triple product can be seen by considering the
parallelepiped determined by the vectors a, b, and c (Figure 3). The area of the base
parallelogram is A
b c . If is the angle between a and b c, then the height h
of the parallelepiped is h
a cos . (We must use cos
instead of cos in case
2.) Therefore, the volume of the parallelepiped is bxc
h¨a
c
b V FIGURE 3 Ah b c a cos a b c Thus, we have proved the following formula.
11 The volume of the parallelepiped determined by the vectors a, b, and c is the
magnitude of their scalar triple product: V a b c If we use the formula in (11) and discover that the volume of the parallelepiped
determined by a, b, and c is 0, then the vectors must lie in the same plane...
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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.
 Fall '09
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