7 b sin in fact that is exactly how corollary two

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 . l l l l 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 l PQ 2 1i 5 4j 1 6k 3i j 7k l PR 1 1i 1 4j 1 6k 5 j 5k We compute the cross product of these vectors: l PR i 3 0 j 1 5 5 l PQ 35 i k 7 5 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 . l l PR 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 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 we obtain i j j k i j k k k i j k i i i k 2, j j Observe that i j j...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online