Unformatted text preview: right-hand rule, and whose length is a
physicists deﬁne a b.
7 b sin . In fact, that is exactly how b 0 Proof Two nonzero vectors a and b are parallel if and only if sin b sin ¨ ¨
FIGURE 2 853 Corollary Two nonzero vectors a and b are parallel if and only if a b ❙❙❙❙ 0, so a b 0 and therefore a b 0 or . In either case 0. The geometric interpretation of Theorem 6 can be seen by looking at Figure 2. If a and
b are represented by directed line segments with the same initial point, then they determine
a parallelogram with base a , altitude b sin , and area
A a a ( b sin ) a b Thus, we have the following way of interpreting the magnitude of a cross product.
The length of the cross product a
determined by a and b. b is equal to the area of the parallelogram EXAMPLE 3 Find a vector perpendicular to the plane that passes through the points P 1, 4, 6 , Q 2, 5, 1 , and R 1, 1, 1 .
SOLUTION The vector PQ
PR is perpendicular to both PQ and PR and is therefore perpendicular to the plane through P, Q, and R. We know from (13.2.1) that
3i j 7k
5 j 5k
We compute the cross product of these vectors:
5 5 l
PQ 35 i k
15 0j 15 0k 40 i 15 j 15 k So the vector
40, 15, 15 is perpendicular to the given plane. Any nonzero scalar
multiple of this vector, such as
8, 3, 3 , is also perpendicular to the plane.
EXAMPLE 4 Find the area of the triangle with vertices P 1, 4, 6 , Q and R 1, 2, 5, 1, 1, 1 .
40, 15, 15 . The area of the
parallelogram with adjacent sides PQ and PR is the length of this cross product: SOLUTION In Example 3 we computed that PQ l
PR s 40 2 15 2 15 2 5s82 The area A of the triangle PQR is half the area of this parallelogram, that is, 5 s82.
2 5E-13(pp 848-857) 854 ❙❙❙❙ 1/18/06 11:21 AM Page 854 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE If we apply Theorems 5 and 6 to the standard basis vectors i , j, and k using
i j j k i j
k k k i j k
i i i
k 2, j
j Observe that
i j j...
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