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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 Fall '09
 hamrick
 Linear Algebra, Vectors, Vector Space, Dot Product, (pp 828837)

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