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Physics LibreTexts

02: Vectors

Check Your Understanding

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

3. Answers may vary

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 \(\hat{k}\)

Additional Problems

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

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