SECTION 9.7 The Cross Product 673 Now Work PROBLEM 49 Solution Two adjacent sides of this parallelogram are = = − + = = − + − PP PP u i j k v i j k 3 2 and 3 1 2 1 3 Since × = − + + u v i j k 2 7 (Example 4), the area of the parallelogram is = × = + + = = u v Area of parallelogram 1 4 49 54 3 6 squareunits. CAUTION Not all pairs of vertices give rise to a side. For example, P P1 4 is a diagonal of the parallelogram since + = P P P P P P . 1 3 3 4 1 4 Also, P P1 3 and P P2 4 are not adjacent sides; they are parallel sides. j Proof of Property (11) The proof requires two parts. If u and v are parallel, then there is a scalar α such that α = u v. Then α α ( ) ( ) × = × = × = u v v v v v 0 ↑ ↑ Property (4) Property (2) If × = u v 0, then, by property (9), we have θ × = = u v u v sin 0 Since ≠ u 0 and ≠ v 0, we must have θ = sin 0, so θ = 0 or θ π = . In either case, since θ is the angle between u and v, then u and v are parallel. ■ Concepts and Vocabulary 9.7 Assess Your Understanding 1. True or False If u and v are parallel vectors, then × = u v 0. 2. True or False For any vector × = v v v 0 , . 3. True or False If u and v are vectors, then × + × = u v v u 0. 4. True or False ×u v is a vector that is parallel to both u and v. 5. True or False θ × = u v u v cos , where θ is the angle between u and v. 6. True or False The area of the parallelogram having u and v as adjacent sides is the magnitude of the cross product of u and v. Skill Building In Problems 7–14, find the value of each determinant. 7. 3 4 1 2 8. − − 2 5 2 3 9. − − 6 5 2 1 10. −4 0 5 3 11. A B C 2 1 4 1 3 1 12. A B C 0 2 4 3 1 3 13. − − A B C 1 3 5 50 2 14. − − − A B C 1 2 3 0 2 2 In Problems 15–22, find (a) ×v w, (b) ×w v, (c) ×w w, and (d) ×v v. 15. = − + v i j k 2 3 = − − w i j k 3 2 16. = − + + v i j k 3 2 = − − w i j k 3 2 17. = + v i j = + + w i j k 2 18. = − + v i j k 4 2 = + + w i j k 3 2 19. = − + v i j k 2 2 = − w j k 20. = + + v i j k 3 3 = − w i k 21. = − − v i j k = − w i k 4 3 22. = − v i j 2 3 = − w j k 3 2 In Problems 23–44, use the given vectors u, v, and w to find each expression. = − + = − + + = + + u i j k v i j k w i j k 2 3 3 3 2 3 23. ×u v 24. ×v w 25. ×v u 26. ×w v 27. ×v v 28. ×w w 29. ( ) × u v 3 30. ( ) ×v w4 31. ( ) ×u v2 32. ( ) − × v w 3 33. ( ) ⋅ × u u v 34. ( ) ⋅ × v v w 35. ( ) ⋅ × u v w 36. ( ) × ⋅ u v w 37. ( ) ⋅ × v u w 38. ( ) × ⋅ v u w 39. ( ) × × u v v 40. ( ) × × w w v 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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