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Unformatted text preview: nces is proj a b proj b a? Determine whether the given vectors are orthogonal,
parallel, or neither.
 23. (a) a 24. (a) u (b) u
(c) u
■ 45. A constant force with vector representation 5, 3, 7 , b
6, 8, 2
4, 6 , b
3, 2
i 2 j 5 k, b 3 i 4 j k
2 i 6 j 4 k, b
3i 9 j 6k (b) a
(c) a
(d) a i ■ 3, 9, 6 , v
j 2 k, v
a, b, c , v ■ ■ F 10 i 18 j 6 k moves an object along a straight line
from the point 2, 3, 0 to the point 4, 9, 15 . Find the work
done if the distance is measured in meters and the magnitude of
the force is measured in newtons. 4 , 12, 8
2i j k
b, a, 0 ■ ■ ■ ■ 46. Find the work done by a force of 20 lb acting in the direction N50 W in moving an object 4 ft due west.
■ ■ ■ ■ pushes it up a ramp that is 10 ft long and inclined at an angle
of 20 above the horizontal. Find the work done on the box. 25. Use vectors to decide whether the triangle with vertices P 1, 3, 2 , Q 2, 0, 4 , and R 6, 2, 5 is rightangled. 48. A wagon is pulled a distance of 100 m along a horizontal path by a constant force of 50 N. The handle of the wagon is held at
an angle of 30 above the horizontal. How much work is done? 2 26. For what values of b are the vectors 6, b, 2 and b, b , b orthogonal?
27. Find a unit vector that is orthogonal to both i 47. A woman exerts a horizontal force of 25 lb on a crate as she j and i 49. Use a scalar projection to show that the distance from a point k. P1 x 1, y1 to the line a x 28. Find two unit vectors that make an angle of 60 with v 29–33  Find the direction cosines and direction angles of the
vector. (Give the direction angles correct to the nearest degree.) 1, 50. If r 2, 31. 2 i
32. 2 i j 33. 6k ■ ■ 0
■ 52. Find the angle between a diagonal of a cube and a diagonal of
■ 34. If a vector has direction angles ■ ■ ■ 4 and ■ ■  Find the scalar and vector projections of b onto a. 35. a 3, 36. a 1, 2 , 4, b
b 5, 0
4, 1 ■ 3, ﬁnd the third direction angle .
35–40 b1, b2 , b3 , show
0 represents a edges. where c ■ 2, 3 to 51. Find...
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 Fall '09
 hamrick

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