45. u = j v=iij 46. u : 2i 2j v = *i i j A Relationship of Two Vectors ln Exercises 4750, determine whether u and v are orthogonal, parallel, or neither. 47. u = (10, 20) 48. u = (15,9) v=(*18,9) v=(*5,*3) 49.u=%i+~175j 50.u=,%~,i+3j v=12il4j v=51+%j Finding an Unknown Vector Component In Exercises 5156, find the value of k such that the vectors 1] and v are orthogonal. Sl.u=2ikj 52.u=8i+4j v=3i+2j v=2iikj 53.u=i+4j 54.u:3ki+5j v=7ki5j v=2i4j SS.u=3ki+2j 56.u:4i4kj v=6i v=3j Decomposing a Vector into Components In Exercises 5760, find the projection of 11 onto v. Then write 11 as the sum of two orthogonal vectors, one of which is projvu. 57. u = (2, 2) 58. u : (4, 2) V: (6,1) V= (1,2> 59. u = (0,3) 60. u = (*5, *1) v:(2, 15) v=(1,l) Finding the Projection of u onto v Mentally ln Exercises 6164, use the graph to determine mentally the projection of u onto v. (The coordinates of the terminal points of the vectors in standard position are given.) Use the formula for the projection of it onto v to verify your result. 61. -" 62. 63. 64. y iStockpiioloLorm-lshotbvdave Section 7.4 Vectors and Dot Products 581 Finding Orthogonal Vectors ln Exercises 6568, nd two vectors in opposite directions that are orthogonal to the vector u. (There are many correct answers.) 65. u = (2, 6) 66. u = (7, 5) 67.u=i3;j 68.u=i3j Finding the Work Done In Exercises 69 and 70, find the work done in moving a particle from P to Q when the magnitude and direction of the force are given by v. 69. P = (0,0), Q = (4,7), v = (1.4) 70. P : (1,3), Q = (3,5), v = 2i + 3j 71. Business The vector u = (1225, 2445) gives the numbers of hours worked by employees of a temp agency at two pay levels. The vector v = (12.00, 10.25) gives the hourly wage (in dollars) paid at each level, respectively. (a) Find the dot product u - v and explain its meaning in the context of the problem. (b) Identify the vector operation used to increase wages by 2 percent. 72. Business The vector u = (3240, 2450) gives the numbers of hamburgers and hot dogs, respectively, sold at a fast food stand in one week. The vector v = (3.25, 3.50) gives the prices in dollars of the food items. (a) Find the dot product 11 - v and explain its meaning in the context of the problem. (b) Identify the vector operation used to increase prices by 2% percent. 73. Wit/vyau 5/rauH/mrnr't (p. 574) A truck with a gross weight of 30,000 pounds is parked on a slope of (1 (see figure). Assume that the only force to overcome is the force of gravity. Weight: 30,000 1b (a) Find the force required to keep the truck from rolling down the hill in terms of the slope d. (b) Use a graphing utility to complete the table. (c) Find the force perpendicular to the hill when d = .5". Vocabulary and Concept Check 1. For two vectors u and v, does u . v = v . u? 2. What is the dot product of two orthogonal vectors? 3. Is the dot product of two vectors an angle, a vector, or a scalar? In Exercises 4-6, fill in the blank(s). 4. If 0 is the angle between two nonzero vectors u and v, then cos 0 = 5. The projection of u onto v is given by projvu = 6. The work W done by a constant force F as its point of application moves along the vector PQ is given by either W = or W = Procedures and Problem Solving Finding the Dot Product In Exercises 7-10, find the dot product of u and v. 30. u = cos Ti+ sin("); 7. u = (6, 3) 8. u = (-4, 1) v = cosm )i + sini V = (2, -4) v = (5, -4) 9. u = 5i + j 10. u = 2i + 6j Finding the Angle Between Two Vectors In Exercises v = 3i - j -3i + 7j 31-34, graph the vectors and find the degree measure of the angle between the vectors. Using Properties of Dot Products In Exercises 11-16, use the vectors u = (2, 2), v = (-5, 3), and w = (1, -4) 31. u = 2i - 4j 32. u = 6i - 2j to find the indicated quantity. State whether the result is v = 3i - 5j v = 8i - 5j a vector or a scalar. 33. u = -6i - 3j 34. u = -7i - 4j 11. u . u 12. v . w V = - 8i + 4j = -8i + 2j 13. u . 2v 14. 4u . V Finding the Angles in a Triangle In Exercises 35-38, use 15. (3w . v)u 16. (u . 2v)w vectors to find the interior angles of the triangle with the Finding the Magnitude of a Vector In Exercises 17-22, given vertices. use the dot product to find the magnitude of u. 35. (1, 2), (3, 4), (2, 5) 36. (-3, -4), (1, 7), (8, 2) 17. u = (-5, 12) 18. u = (-8, 15) 37. (-3, 0), (2, 2), (0, 6) 38. (-3, 5), (- 1, 9), (7, 9) 19. u = 201 + 25j 20. u = 6i - 10j Using the Angle Between Two Vectors In Exercises 21. u = -4j 22. u = 9i 39-42, find u . v, where 0 is the angle between u and v. Finding the Angle Between Two Vectors In Exercises 39. lull = 9, 1/v/| = 36, 0 = : 23-30, find the angle 0 between the vectors. 4 23. u = (-1, 0) 24. u = (4, 4) 40. |ull = 4, |/v/| = 12, 0 = 1 V = (0, 2) V = (-2, 0) 25. u = 2i + 6j 26. u = 7i - 2j 41. |ull = 4, ||v/| = 10, 0 = 2 Tt 3 v = -5i + 2j v = -8i + 6j 27. u = 2i 28. u = 4j 42. Hull = 100, ||v/| = 250, 0 = v = -3j v = -9i 29. u = cos Zi+ sin("); Determining Orthogonal Vectors In Exercises 43-46, determine whether u and v are orthogonal. V = cos 3Tti + sin 43. u = (10, -6) 44. u = (12, 4) V = (9, 15) v = (4, -3) iStockphoto.com/shotbydave74. Physics A sport utility vehicle with a gross weight of 5400 pounds is parked on a slope of 10. Assume that the only force to overcome is the force of gravity. Find the force required to keep the vehicle from rolling down the hill. Find the force perpendicular to the hill. 7S. MODELING DATA ' One of the events in a local strongman contest is to drag a concrete block. One competitor drags the block with a constant force of 250 pounds at a constant angle of 30 with the horizontal. (See figure.) (a) Find the work done in terms of the distance d. 030 Use a graphing utility to complete the table. a' (in feet) Work 76. Public Safety A ski patroller pulls a rescue toboggan across a at snow surface by exerting a constant force of 35 pounds on a handle that makes a constant angle of 22 with the horizontal (see figure). Find the work done in pulling the toboggan 200 feet. 77. Work A force of 50 pounds, exerted at an angle of 25 with the horizontal, is required to slide a desk across a floor. Determine the work done in sliding the desk 15 feet. 78. Work A mover exerts a horizontal force of 25 pounds on a crate as it is pushed up a ramp that is 12 feet long and inclined at an angle of 20 above the horizontal. Find the work done in pushing the crate up the ramp. Conclusions True or False? In Exercises 79 and 80, determine whether the statement is true or false. Justify your answer. 79. The vectors u 2 (0, 0) and v 2 ( l2, 6) are orthogonal. 80. The work W done by a constant force F acting along the line of motion of an object is represented by a vector. 81. Think About It If u = (cos 6, sin 9) and v = (sin 6', icos 6), are u and v orthogonal, parallel, or neither? Explain. 82. Error Analysis Describe the error. nal-a Hula-um 83. Think About It Let 11 be a unit vector. What is the value of u - 1.1? Explain. H ""\\___ % HOW DO YOU SEE IT? What is known about 6, the angle between two nonzero vectors 1! and v (see gure) under each condition? 11 6 v Origin (a)u-v=0 (b)u-v>0 (c)u-v