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2.A: Vectors (Answers)

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    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; \(\vec{D}=−16m\hat{u}\)

    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

    3. answers may vary

    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. \(\vec{F}=−17.32\hat{i}−10.00\hat{j}\)

    The x y coordinate system is shown, with positive x to the right and positive y up. Vector A has magnitude 10.0 and makes an angle of 30 degrees above the positive x direction. Vector B has magnitude 5.0 and makes an angle of 53 degrees above the positive x direction. Vector C has magnitude 12.0 and makes an angle of 60 degrees below the positive x direction. Vector D has magnitude 20.0 and makes an angle of 37 degrees above the negative x direction. Vector F has magnitude 20.0 and makes an angle of 30 degrees below the negative x direction.

    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\(\hat{k}\)

    Additional Problems

    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

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