658 CHAPTER 9 Polar Coordinates; Vectors In Problems 21–26, decompose v into two vectors v1 and v2 , where v1 is parallel to w, and v2 is orthogonal to w. 21. v i j w i j 2 3 , = − = − 22. v i j w i j 3 2 , 2 = − + = + 23. v i j w i j , 2 = − = − − 24. v i j w i j 2 , 2 = − = − 25. v i j w i j 3 , 2 = + = − − 26. v i j w i j 3 , 4 = − = − 27. Find a vector of magnitude 15 that is parallel to i j 4 3 − . 28. Find a vector of magnitude 5 that is parallel to i j 12 9 . − + 29. Computing Work Find the work done by a force of 3 pounds acting in a direction of 60° to the horizontal in moving an object 6 feet from 0, 0 ( ) to 6, 0 ( ). 30. Computing Work A wagon is pulled horizontally by exerting a force of 20 pounds on the handle at an angle of 30° with the horizontal. How much work is done in moving the wagon 100 feet? 31. Solar Energy The amount of energy collected by a solar panel depends on the intensity of the sun’s rays and the area of the panel. Let the vector I represent the intensity, in watts per square centimeter, having the direction of the sun’s rays. Let the vector A represent the area, in square centimeters, whose direction is the orientation of a solar panel. See the figure.The total number of watts collected by the panel is given by W A I . = ⋅ A I Suppose that = 〈− − 〉 I 0.02, 0.01 and A 300, 400. = 〈 〉 (a) Find I and A , and interpret the meaning of each. (b) Compute W and interpret its meaning. (c) If the solar panel is to collect the maximum number of watts, what must be true about I and A? 32. Rainfall Measurement Let the vector R represent the amount of rainfall, in inches, whose direction is the inclination of the rain to a rain gauge. Let the vector A represent the area, in square inches, whose direction is the orientation of the opening of the rain gauge. See the figure. The volume of rain collected in the gauge, in cubic inches, is given by V R A, = ⋅ even when the rain falls in a slanted direction or the gauge is not perfectly vertical. A R 9 8 7 6 5 4 3 2 1 Suppose that = 〈 − 〉 R 0.75, 1.75 and A 0.3, 1. = 〈 〉 (a) Find R and A , and interpret the meaning of each. (b) Compute V and interpret its meaning. (c) If the gauge is to collect the maximum volume of rain, what must be true about R and A? 33. Braking Load A Ford Explorer with a gross weight of 5300 pounds is parked on a street with an 8° grade. See the figure. Find the magnitude of the force required to keep the Explorer from rolling down the hill. What is the magnitude of the force perpendicular to the hill? Weight 5 5300 pounds 88 34. Braking Load A Chevrolet Silverado with a gross weight of 4500 pounds is parked on a street with a 10° grade. Find the magnitude of the force required to keep the Silverado from rolling down the hill.What is the magnitude of the force perpendicular to the hill? 35. Ramp Angle Billy and Timmy are using a ramp to load furniture into a truck. While rolling a 250-pound piano up the ramp, they discover that the truck is too full of other furniture for the piano to fit. Timmy holds the piano in place on the ramp while Billy repositions other items to make room for it in the truck. If the angle of inclination of the ramp is 20°, how many pounds of force must Timmy exert to hold the piano in position? 208 250 lb 36. Incline Angle A bulldozer exerts 1000 pounds of force to prevent a 5000-pound boulder from rolling down a hill. Determine the angle of inclination of the hill. 37. Find the acute angle that a constant unit force vector makes with the positive x-axis if the work done by the force in moving a particle from 0, 0 ( ) to 4, 0 ( ) equals 2. 38. Prove the distributive property: u v w u v u w ( ) ⋅ + = ⋅ + ⋅ 39. Prove property (5): ⋅ = 0 v 0. 40. If v is a unit vector and the angle between v and i is ,α show that v i j cos sin . α α = + 41. Suppose that v and w are unit vectors. If the angle between v and i is α and the angle between w and i is ,β use the idea of the dot product v w⋅ to prove that cos cos cos sin sin α β α β α β ( ) − = + Applications and Extensions
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