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31 a woman walks due west on the deck of a ship at 3

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Unformatted text preview: e and the wind. The ground speed of the plane is the magnitude of the resultant. Find the true course and the ground speed of the plane. 31. A woman walks due west on the deck of a ship at 3 mi h. The ship is moving north at a speed of 22 mi h. Find the speed and direction of the woman relative to the surface of the water. 32. Ropes 3 m and 5 m in length are fastened to a holiday decora- tion that is suspended over a town square. The decoration has a mass of 5 kg. The ropes, fastened at different heights, make angles of 52 and 40 with the horizontal. Find the tension in each wire and the magnitude of each tension. 40° 52° 38. Suppose that a and b are nonzero vectors that are not parallel and c is any vector in the plane determined by a and b. Give a geometric argument to show that c can be written as c s a t b for suitable scalars s and t. Then give an argument using components. x, y, z and r0 x 0 , y0 , z0 , describe the set of all points x, y, z such that r r0 1. 39. If r 40. If r x, y , r1 x 1, y1 , and r2 set of all points x, y such that r where k r1 r2 . r1 x 2 , y2 , describe the r r2 k, 41. Figure 16 gives a geometric demonstration of Property 2 of vectors. Use components to give an algebraic proof of this fact for the case n 2. 42. Prove Property 5 of vectors algebraically for the case n 3. Then use similar triangles to give a geometric proof. 5 m 3 m 3, 2 , b 2, 1 , and c 7, 1 . (b) Show, by means of a sketch, that there are scalars s and t such that c s a t b. (c) Use the sketch to estimate the values of s and t. (d) Find the exact values of s and t. 43. Use vectors to prove that the line joining the midpoints of two sides of a triangle is parallel to the third side and half its length. 44. Suppose the three coordinate planes are all mirrored and a 33. A clothesline is tied between two poles, 8 m apart. The line is quite taut and has negligible sag. When a wet shirt with a mass of 0.8 kg is hung at the middle of the line, the midpoint is pulled down 8 cm. Find the tension in...
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