807 8.3 Geometrically Defined Vectors and Applications 8.3 Exercises CONCEPT PREVIEW Refer to the vectors m through t below. 1. Name all pairs of vectors that appear to be equal. 2. Name all pairs of vectors that are opposites. 3. Name all pairs of vectors where the first is a scalar multiple of the other, with the scalar positive. 4. Name all pairs of vectors where the first is a scalar multiple of the other, with the scalar negative. n m o r s t p q CONCEPT PREVIEW Refer to vectors a through h below. Make a copy or a sketch of each vector, and then draw a sketch to represent each of the following. For example, find a + e by placing a and e so that their initial points coincide. Then use the parallelogram rule to find the resultant, as shown in the figure on the right. b d f e g h a c e a a + e 5. -b 6. -g 7. 2c 8. 2h 9. a + b 10. h + g 11. a - c 12. d - e 13. a + 1b + c2 14. 1a + b2 + c 15. c + d 16. d + c 17. From the results of Exercises 13 and 14, does it appear that vector addition is associative? 18. From the results of Exercises 15 and 16, does it appear that vector addition is commutative? For each pair of vectors u and v with angle u between them, sketch the resultant. 19. u = 12, v = 20, u = 27° 20. u = 8, v = 12, u = 20° 21. u = 20, v = 30, u = 30° 22. u = 50, v = 70, u = 40° Use the parallelogram rule to find the magnitude of the resultant force for the two forces shown in each figure. Round answers to the nearest tenth. 23. 24. 40 lb 60 lb 408 85 lb 102 lb 658 25. 26. 15 lb 25 lb 1108 1500 lb 2000 lb 1408
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