Solutions: Vectors

Solutions to Try Its

1.
A vector from the origin to (3,5) - a line with an arrow at the (3,5) endpoint.

2. [latex]3u=\langle 15,12\rangle [/latex]

3. [latex]u=8i - 11j[/latex]

4. [latex]v=\sqrt{34}\cos \left(59^\circ \right)i+\sqrt{34}\sin \left(59^\circ \right)j[/latex]
Magnitude = [latex]\sqrt{34}[/latex]
[latex]\theta ={\tan }^{-1}\left(\frac{5}{3}\right)=59.04^\circ [/latex]

Solutions to Odd-Numbered Exercises

1. lowercase, bold letter, usually u, v, w

3. They are unit vectors. They are used to represent the horizontal and vertical components of a vector. They each have a magnitude of 1.

5. The first number always represents the coefficient of the i, and the second represents the j.

7. [latex]\langle 7,−5\rangle[/latex]

9. not equal

11. equal

13. equal

15. [latex]7i−3j[/latex]

17. [latex]−6i−2j[/latex]

19. [latex]u+v=\langle−5,5\rangle,u−v=\langle−1,3\rangle,2u−3v=\langle 0,5\rangle[/latex]

21. [latex]−10i–4j[/latex]

23. [latex]−\frac{2\sqrt{29}}{29}i+\frac{5\sqrt{29}}{29}j[/latex]

25. [latex]–\frac{2\sqrt{229}}{229}i+\frac{15\sqrt{229}}{229}j[/latex]

27. [latex]–\frac{7\sqrt{2}}i+\frac{\sqrt{2}}{10}j[/latex]

29. [latex]|v|=7.810,\theta=39.806^{\circ}[/latex]

31. [latex]|v|=7.211,\theta=236.310^{\circ}[/latex]

33. −6

35. −12

37.

39.
Plot of u+v, u-v, and 2u based on the above vectors. In relation to the same origin point, u+v goes to (0,3), u-v goes to (2,-1), and 2u goes to (2,2).

41.
Plot of vectors u+v, u-v, and 2u based on the above vectors.Given that u's start point was the origin, u+v starts at the origin and goes to (2,-3); u-v starts at the origin and goes to (4,-1); 2u goes from the origin to (6,-4).

43.
Plot of a single vector. Taking the start point of the vector as (0,0) from the above set up, the vector goes from the origin to (-1,-6).

45.
Vector extending from the origin to (7,5), taking the base as the origin.

47. [latex]\langle 4,1\rangle[/latex]

49. [latex]v=−7i+3j[/latex]
Vector going from (4,-1) to (-3,2).

51. [latex]3\sqrt{2}i+3\sqrt{2}j[/latex]

53. [latex]i−\sqrt{3}j[/latex]

55. a. 58.7; b. 12.5

57. [latex]x=7.13[/latex] pounds, [latex]y=3.63[/latex] pounds

59. [latex]x=2.87[/latex] pounds, [latex]y=4.10[/latex] pounds

61. 4.635 miles, [latex]17.764^{\circ}[/latex] N of E

63. 17 miles. 10.318 miles

65. Distance: 2.868. Direction: [latex]86.474^{\circ}[/latex] North of West, or [latex]3.526^{\circ}[/latex] West of North

67. [latex]4.924^{\circ}[/latex]. 659 km/hr

69. [latex]4.424^{\circ}[/latex]

71. (0.081, 8.602)

73. [latex]21.801^{\circ}[/latex], relative to the car’s forward direction

75. parallel: 16.28, perpendicular: 47.28 pounds

77. 19.35 pounds, [latex]231.54^{\circ}[/latex] from the horizontal

79. 5.1583 pounds, [latex]75.8^{\circ}[/latex] from the horizontal