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• • OpenStax
• General Physics at OpenStax CNX

2.1. a. not equal because they are orthogonal;

b. not equal because they have different magnitudes;

c. not equal because they have different magnitudes and directions;

d. not equal because they are antiparallel;

e. equal.

2.2. 16 m; $$\vec{D}=−16m\hat{u}$$

2.3. G = 28.2 cm, $$θ_G=291°$$

2.4. $$\vec{D}=(−5.0\hat{i}−3.0\hat{j})cm$$; the fly moved 5.0 cm to the left and 3.0 cm down from its landing site.

2.5. 5.83 cm, $$211°$$

2.6. $$\vec{D}=(−20m)\hat{j}$$

2.7. 35.1 m/s = 126.4 km/h

2.8. $$\vec{G}=(10.25\hat{i}−26.22\hat{j})cm$$

2.9. D = 55.7 N; direction $$65.7°$$ north of east

2.10. $$\hat{v}=0.8\hat{i}+0.6\hat{j}, 36.87°$$ north of east

2.11. $$\vec{A}⋅\vec{B}=−57.3, \vec{F}⋅\vec{C}=27.8$$

2.13. $$131.9°$$

2.14. $$W_1=1.5J, W_2=0.3J$$

2.15. $$\vec{A}×\vec{B}=−40.1\hat{k}$$ or, equivalently, $$∣\vec{A}×\vec{B}∣=40.1$$, and the direction is into the page; $$\vec{C}×\vec{F}=+157.6\hat{k}$$ or, equivalently, $$∣\vec{C}×\vec{F}∣=157.6$$, and the direction is out of the page.

2.16. a. $$−2\hat{k}$$,

b. 2,

c. $$153.4°$$,

d. $$135°$$

## Conceptual Questions

1. scalar

5. parallel, sum of magnitudes, antiparallel, zero

7. no, yes

9. zero, yes

11. no

13. equal, equal, the same

15. a unit vector of the x-axis

17. They are equal.

19. yes

21. a. $$C=\vec{A}⋅\vec{B}$$

b. $$\vec{C}=\vec{A}·\vec{B}$$ or $$\vec{C}=\vec{A}−\vec{B}$$

c. $$\vec{C}=\vec{A}×\vec{B}$$,

d. $$\vec{C}=A\vec{B}$$,

e. $$\vec{C}+2\vec{A}=\vec{B}$$,

f. $$\vec{C}=\vec{A}×\vec{B}$$,

g. left side is a scalar and right side is a vector,

h. $$\vec{C}=2\vec{A}×\vec{B}$$,

i. $$\vec{C}=\vec{A}/B$$,

j. $$\vec{C}=\vec{A}/B$$

23. They are orthogonal.

## Problems

25. $$\vec{h}=−49m\hat{u}$$, 49 m

27. 30.8 m, $$35.7°$$ west of north

29. 134 km, $$80°$$

31. 7.34 km, $$63.5°$$ south of east

33. 3.8 km east, 3.2 km north, 7.0 km

35. 14.3 km, $$65°$$

37. a. $$\vec{A}=+8.66\hat{i}+5.00\hat{j}$$,

b. $$\vec{B}=+3.01\hat{i}+3.99\hat{j}$$,

c. $$\vec{C}=+6.00\hat{i}−10.39\hat{j}$$,

d. $$\vec{D⃗}=−15.97\hat{i}+12.04\hat{j}$$,

f. $$\vec{F}=−17.32\hat{i}−10.00\hat{j}$$ 39. a. 1.94 km, 7.24 km;

b. proof

41. 3.8 km east, 3.2 km north, 2.0 km, $$\vec{D}=(3.8\hat{i}+3.2\hat{j})km 43. \(P_1(2.165m,1.250m), P_2(−1.900m,3.290m), 5.27 m$$

45. 8.60 m, $$A(2\sqrt{5}m,0.647π), B(3\sqrt{2}m,0.75π)$$

47. a. $$\vec{A}+\vec{B}=−4\hat{i}−6\hat{j}$$, $$|\vec{A}+\vec{B}∣=7.211,θ=236.3°$$;

b. $$\vec{A}−\vec{B}=-2\hat{i}+2\hat{j}, ∣\vec{A}−\vec{B}∣=2\sqrt{2},θ=135°$$

49. a. $$\vec{C}=(5.0\hat{i}−1.0\hat{j}−3.0\hat{k})m,C=5.92m$$;

b. $$\vec{D}=(4.0\hat{i}−11.0\hat{j}+15.0\hat{k})m,D=19.03m$$.

51. $$\vec{D}=(3.3\hat{i}−6.6\hat{j})km, \hat{i} is to the east, 7.34 km, −63.5°$$

53. a.$$\vec{R}=−1.35\hat{i}−22.04\hat{j}$$,

b. $$\vec{R}=−17.98\hat{i}+0.89\hat{j}$$

55. $$\vec{D}=(200\hat{i}+300\hat{j})yd$$, D = 360.5 yd, $$56.3°$$ north of east; The numerical answers would stay the same but the physical unit would be meters. The physical meaning and distances would be about the same because 1 yd is comparable with 1 m.

57. $$\vec{R}=−3\hat{i}−16\hat{j}$$

59. $$\vec{E⃗}=E\hat{E}, E_x=+178.9V/m, E_y=−357.8V/m, E_z=0.0V/m, θ_E=−tan^{−1}(2)$$

61. a. $$\vec{R}_B=(12.278\hat{i}+7.089\hat{j}+2.500\hat{k})km, \(\vec{R}_D=(−0.262\hat{i}+3.000\hat{k})km$$;

b. $$∣\vec{R}_B−\vec{R}_D∣=14.414km$$|R→B−R→D|=14.414km

63. a. 8.66,

b. 10.39,

c. 0.866,

d. 17.32

65. $$θ_i=64.12°,θ_j=150.79°,θ_k=77.39°$$

67. a. $$−119.98\hat{k}$$

b. $$0\hat{k}$$,

c. $$+93.69\hat{k}$$,

d. $$−240.0\hat{k}$$,

e. $$+3.993\hat{k}$$,

f. $$−3.009\hat{k}$$,

g. $$+14.99\hat{k}$$,

h. 0

69. a. 0,

b. 173,194,

c. +199,993$$\hat{k}$$

71. a. 18.4 km and 26.2 km,

b. 31.5 km and 5.56 km

73. a. $$(r,φ+π/2)$$,

b. $$(2r,φ+2π)($$,

c. $$(3r,−φ)$$

75. $$d_{PM}=33.12nmi=61.34km,d_{NP}=35.47nmi=65.69km$$

77. proof

79. a. 10.00 m,

b. $$5πm$$,

c. 0

81. 22.2 km/h, $$35.8°$$. south of west

83. 240.2 m, $$2.2°$$ south of west

85. $$\vec{B}=−4.0\hat{i}+3.0\hat{j}$$ or $$\vec{B}=4.0\hat{i}−3.0\hat{j}$$

87. proof

## Challenge Problems

89. $$G_{⊥}=2375\sqrt{17}≈9792$$

91. proof