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# 02: Vectors

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; $$\vec{D}$$ = −16 m $$\hat{u}$$

2.3. G = 28.2 cm, $$\theta_{G}$$ = 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. $$\vec{D}$$ = (−20 m) $$\hat{j}$$

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

2.13. 131.9°

2.14. W1 = 1.5 J, W2 = 0.3 J

2.15. $$\vec{A} \times \vec{B}$$ = −40.1 $$\hat{k}$$ or, equivalently, |$$\vec{A} \times \vec{B}$$| = 40.1, and the direction is into the page; $$\vec{C} \times \vec{F}$$ = + 157.6 $$\hat{k}$$ or, equivalently, |$$\vec{C} \times \vec{F}$$| = 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. Yes, 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 = $$\vec{A} \cdotp \vec{B}$$, b. $$\vec{C} = \vec{A} \times \vec{B}$$ or $$\vec{C} = \vec{A} - \vec{B}$$, c. $$\vec{C} = \vec{A} \times\vec{B}$$, d. $$\vec{C}$$ = A$$\vec{B}$$, e. $$\vec{C} + 2 \vec{A} = \vec{B}$$, f. $$\vec{C} = \vec{A} \times \vec{B}$$, g. left side is a scalar and right side is a vector, h. $$\vec{C} = 2 \vec{A} \times \vec{B}$$, i. $$\vec{C} = \frac{\vec{A}}{B}$$, j. $$\vec{C} = \frac{\vec{A}}{B}$$

23. They are orthogonal.

# Problems

25. $$\vec{h}$$ = −16.4 m $$\hat{u}$$, 16.4 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. $$\vec{B}$$ = + 30.09 $$\hat{i}$$ + 39.93 $$\hat{j}$$

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

d. $$\vec{D}$$ = −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. P1(2.165 m, 1.250 m), P2(−1.900 m, 3.290 m), 5.27 m

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

47. a. $$\vec{A} + \vec{B}$$ = −4 $$\hat{i}$$ − 6 $$\hat{j}$$, |$$\vec{A} + \vec{B}$$| = 7.211, $$\theta$$ = 213.7°

b. $$\vec{A} -\vec{B}$$ = 2 $$\hat{i}$$ − 2 $$\hat{j}$$, |$$\vec{A} - \vec{B}$$| = 2$$\sqrt{2}$$, $$\theta$$ = −45°

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

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

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. $$\vec{E}$$ = E $$\hat{E}$$, Ex = + 178.9 V/m , Ey = −357.8 V/m, Ez = 0.0 V/m, $$\theta_{E}$$ = −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. |$$\vec{R}_{B} − \vec{R}_{D}$$| = 14.414 km

63. a. 8.66

b. 10.39

c. 0.866

d. 17.32

65. $$\theta_{i}$$ = 64.12°, $$\theta_{j}$$ = 150.79°, $$\theta_{k}$$ = 77.39°

67. a. −119.98 $$\hat{k}$$

b. −173.2 $$\hat{k}$$

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

d. −413.2 $$\hat{k}$$

e. +39.93 $$\hat{k}$$

f. −30.09 $$\hat{k}$$

g. +149.9 $$\hat{k}$$

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, $$\phi + \frac{\pi}{2}$$)

b. (2r, $$\phi + 2 \pi$$)

c. (3r, −$$\phi$$)

75. dPM = 33.12 nmi = 61.34 km, dNP = 35.47 nmi = 65.69 km

77. proof

79. a. 10.00 m

b. 5$$\pi$$ m, 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$$\perp$$ = 2375$$\sqrt{17}$$ ≈ 9792

91. proof

# Contributors

Samuel J. Ling (Truman State University), Jeff Sanny (Loyola Marymount University), and Bill Moebs with many contributing authors. This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0).