"

Section 9.1 Answers

  1. .
    • [latex]\vec{v} + \vec{w} = \left\langle 15,-1 \right\rangle[/latex], vector
    • [latex]\vec{w} - 2\vec{v} = \left\langle-21,14 \right\rangle[/latex], vector
    • [latex]\| \vec{v} + \vec{w} \| = \sqrt{226}[/latex], scalar
    • [latex]\| \vec{v} \| + \| \vec{w}\| = 18[/latex], scalar
    • [latex]\| \vec{v} \| \vec{w} - \| \vec{w} \| \vec{v} = \left\langle-21,77\right\rangle[/latex], vector
    • [latex]\|w\| \hat{v}= \left\langle\frac{60}{13}, -\frac{25}{13} \right\rangle[/latex], vector
  2. .
    • [latex]\vec{v} + \vec{w} = \left\langle-12,12 \right\rangle[/latex], vector
    • [latex]\vec{w}  - 2\vec{v}  = \left\langle9,-60 \right\rangle[/latex], vector
    • [latex]\| \vec{v} + \vec{w} \| = 12\sqrt{2}[/latex], scalar
    • [latex]\| \vec{v} \| + \| \vec{w}\| = 38[/latex], scalar
    • [latex]\| \vec{v} \| \vec{w} - \| \vec{w} \| \vec{v}  = \left\langle-34,-612\right\rangle[/latex], vector
    • [latex]\|w\| \hat{v}= \left\langle-\frac{91}{25}, \frac{312}{25} \right\rangle[/latex], vector
  3. .
    • [latex]\vec{v} + \vec{w} = \left\langle0,3\right\rangle[/latex], vector
    • [latex]\vec{w}  - 2\vec{v}  = \left\langle-6,6 \right\rangle[/latex], vector
    • [latex]\| \vec{v} + \vec{w} \| = 3[/latex], scalar
    • [latex]\| \vec{v} \| + \| \vec{w}\| = 3\sqrt{5}[/latex], scalar
    • [latex]\| \vec{v} \| \vec{w} - \| \vec{w} \| \vec{v}  = \left\langle-6\sqrt{5},6\sqrt{5}\right\rangle[/latex], vector
    • [latex]\|w\| \hat{v}= \left\langle4, -2 \right\rangle[/latex], vector
  4. .
    • [latex]\vec{v} + \vec{w} = \left\langle8,9\right\rangle[/latex], vector
    • [latex]\vec{w}  - 2\vec{v}  = \left\langle-22, -3 \right\rangle[/latex], vector
    • [latex]\| \vec{v} + \vec{w} \| = \sqrt{145}[/latex], scalar
    • [latex]\| \vec{v} \| + \| \vec{w}\| = 3\sqrt{29}[/latex], scalar
    • [latex]\| \vec{v} \| \vec{w} - \| \vec{w} \| \vec{v}  = \left\langle-14\sqrt{29},6\sqrt{29}\right\rangle[/latex], vector
    • [latex]\|w\| \hat{v}= \left\langle5, 2 \right\rangle[/latex], vector
  5. .
    • [latex]\vec{v} + \vec{w} = \left\langle\sqrt{3},3\right\rangle[/latex], vector
    • [latex]\vec{w}  - 2\vec{v}  = \left\langle4\sqrt{3}, 0 \right\rangle[/latex], vector
    • [latex]\| \vec{v} + \vec{w} \| = 2\sqrt{3}[/latex], scalar
    • [latex]\| \vec{v} \| + \| \vec{w}\| = 6[/latex], scalar
    • [latex]\| \vec{v} \| \vec{w} - \| \vec{w} \| \vec{v}  = \left\langle8\sqrt{3},0\right\rangle[/latex], vector
    • [latex]\|w\| \hat{v}= \left\langle-2\sqrt{3}, 2 \right\rangle[/latex], vector
  6. .
    • [latex]\vec{v} + \vec{w} = \left\langle-\frac{1}{5},\frac{7}{5}\right\rangle[/latex], vector
    • [latex]\vec{w}  - 2\vec{v}  = \left\langle-2, -1 \right\rangle[/latex], vector
    • [latex]\| \vec{v} + \vec{w} \| = \sqrt{2}[/latex], scalar
    • [latex]\| \vec{v} \| + \| \vec{w}\| = 2[/latex], scalar
    • [latex]\| \vec{v} \| \vec{w} - \| \vec{w} \| \vec{v}  = \left\langle-\frac{7}{5},-\frac{1}{5}\right\rangle[/latex], vector
    • [latex]\|w\| \hat{v}= \left\langle\frac{3}{5}, \frac{4}{5} \right\rangle[/latex], vector
  7. .
    • [latex]\vec{v} + \vec{w} = \left\langle0,0\right\rangle[/latex], vector
    • [latex]\vec{w}  - 2\vec{v}  = \left\langle-\frac{3\sqrt{2}}{2}, \frac{3\sqrt{2}}{2} \right\rangle[/latex], vector
    • [latex]\| \vec{v} + \vec{w} \| = 0[/latex], scalar
    • [latex]\| \vec{v} \| + \| \vec{w}\| = 2[/latex], scalar
    • [latex]\| \vec{v} \| \vec{w} - \| \vec{w} \| \vec{v}  = \left\langle-\sqrt{2},\sqrt{2}\right\rangle[/latex], vector
    • [latex]\|w\| \hat{v}= \left\langle\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \right\rangle[/latex], vector
  8. .
    • [latex]\vec{v} + \vec{w} = \left\langle-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right\rangle[/latex], vector
    • [latex]\vec{w}  - 2\vec{v}  = \left\langle-2, -2\sqrt{3} \right\rangle[/latex], vector
    • [latex]\| \vec{v} + \vec{w} \| = 1[/latex], scalar
    • [latex]\| \vec{v} \| + \| \vec{w}\| = 3[/latex], scalar
    • [latex]\| \vec{v} \| \vec{w} - \| \vec{w} \| \vec{v}  = \left\langle-2,-2\sqrt{3}\right\rangle[/latex], vector
    • [latex]\|w\| \hat{v}= \left\langle1, \sqrt{3} \right\rangle[/latex], vector
  9. .
    • [latex]\vec{v} + \vec{w} = \left\langle3,2\right\rangle[/latex], vector
    • [latex]\vec{w}  - 2\vec{v}  = \left\langle-6, -10 \right\rangle[/latex], vector
    • [latex]\| \vec{v} + \vec{w} \| = \sqrt{13}[/latex], scalar
    • [latex]\| \vec{v} \| + \| \vec{w}\| = 7[/latex], scalar
    • [latex]\| \vec{v} \| \vec{w} - \| \vec{w} \| \vec{v}  = \left\langle-6,-18\right\rangle[/latex], vector
    • [latex]\|w\| \hat{v}= \left\langle\frac{6}{5}, \frac{8}{5}\right\rangle[/latex], vector
  10. .
    • [latex]\vec{v} + \vec{w} = \left\langle1,0\right\rangle[/latex], vector
    • [latex]\vec{w}  - 2\vec{v}  = \left\langle-\frac{1}{2}, -\frac{3}{2} \right\rangle[/latex], vector
    • [latex]\| \vec{v} + \vec{w} \| = 1[/latex], scalar
    • [latex]\| \vec{v} \| + \| \vec{w}\| = \sqrt{2}[/latex], scalar
    • [latex]\| \vec{v} \| \vec{w} - \| \vec{w} \| \vec{v}  = \left\langle0,-\frac{\sqrt{2}}{2}\right\rangle[/latex], vector
    • [latex]\|w\| \hat{v}= \left\langle\frac{1}{2}, \frac{1}{2}\right\rangle[/latex], vector
  11. [latex]\vec{v} = \left\langle3,3\sqrt{3}\right\rangle[/latex]
  12. [latex]\vec{v} = \left\langle\frac{3\sqrt{2}}{2},\frac{3\sqrt{2}}{2}\right\rangle[/latex]
  13. [latex]\vec{v} = \left\langle \frac{\sqrt{3}}{3}, \frac{1}{3}\right\rangle[/latex]
  14. [latex]\vec{v} = \left\langle0,12\right\rangle[/latex]
  15. [latex]\vec{v} = \left\langle-2\sqrt{3}, 2\right\rangle[/latex]
  16. [latex]\vec{v} = \left\langle-\sqrt{3}, 3\right\rangle[/latex]
  17. [latex]\vec{v} = \left\langle-\frac{7}{2}, 0\right\rangle[/latex]
  18. [latex]\vec{v} = \left\langle-5\sqrt{3}, -5\sqrt{3}\right\rangle[/latex]
  19. [latex]\vec{v} = \left\langle0, -6.25\right\rangle[/latex]
  20. [latex]\vec{v} = \left\langle6, -2\sqrt{3}\right\rangle[/latex]
  21. [latex]\vec{v} = \left\langle5, -5\right\rangle[/latex]
  22. [latex]\vec{v} = \left\langle2,4\right\rangle[/latex]
  23. [latex]\vec{v} = \left\langle-1, 3\right\rangle[/latex]
  24. [latex]\vec{v} = \left\langle-3, -4\right\rangle[/latex]
  25. [latex]\vec{v} = \left\langle24, -10\right\rangle[/latex]
  26. [latex]\vec{v} \approx \left\langle-177.96, 349.27\right\rangle[/latex]
  27. [latex]\vec{v} \approx \left\langle12.96, 62.59\right\rangle[/latex]
  28. [latex]\vec{v} \approx \left\langle5164.62, 1097.77\right\rangle[/latex]
  29. [latex]\vec{v} \approx \left\langle-386.73, -230.08\right\rangle[/latex]
  30. [latex]\vec{v} \approx \left\langle-52.13, -160.44\right\rangle[/latex]
  31. [latex]\vec{v} \approx \left\langle14.73, -21.43\right\rangle[/latex]
  32. [latex]\|\vec{v}\| = 2[/latex], [latex]\theta = 60^{\circ}[/latex]
  33. [latex]\|\vec{v}\| = 5\sqrt{2}[/latex], [latex]\theta = 45^{\circ}[/latex]
  34. [latex]\|\vec{v}\| = 4[/latex], [latex]\theta = 150^{\circ}[/latex]
  35. [latex]\|\vec{v}\| = 2[/latex], [latex]\theta = 135^{\circ}[/latex]
  36. [latex]\|\vec{v}\| = 1[/latex], [latex]\theta = 225^{\circ}[/latex]
  37. [latex]\|\vec{v}\| = 1[/latex], [latex]\theta = 240^{\circ}[/latex]
  38. [latex]\|\vec{v}\| = 6[/latex], [latex]\theta = 0^{\circ}[/latex]
  39. [latex]\|\vec{v}\| = 2.5[/latex], [latex]\theta = 180^{\circ}[/latex]
  40. [latex]\|\vec{v}\| = \sqrt{7}[/latex], [latex]\theta = 90^{\circ}[/latex]
  41. [latex]\|\vec{v}\| = 10[/latex], [latex]\theta = 270^{\circ}[/latex]
  42. [latex]\|\vec{v}\| = 5[/latex], \\ [latex]\theta = \arctan \left( \frac{4}{3} \right) \approx 53.13^{\circ}[/latex]
  43. [latex]\|\vec{v}\| = 13[/latex],\\ [latex]\theta = \arctan \left( \frac{5}{12} \right) \approx 22.62^{\circ}[/latex]
  44. [latex]\|\vec{v}\| = 5[/latex], \\ [latex]\theta = \arctan \left( -\frac{3}{4} \right) \approx 143.13^{\circ}[/latex]
  45. [latex]\|\vec{v}\| = 25[/latex], \\ [latex]\theta = \arctan \left(- \frac{24}{7} \right)\approx 106.26^{\circ}[/latex]
  46. [latex]\|\vec{v}\| = \sqrt{5}[/latex], \\ [latex]\theta = \arctan \left( \frac{1}{2} \right) + \pi \approx 206.57^{\circ}[/latex]
  47. [latex]\|\vec{v}\| = 2\sqrt{10}[/latex], \\ [latex]\theta = \arctan \left( 3 \right) + \pi \approx 251.57^{\circ}[/latex]
  48. [latex]\|\vec{v}\| = \sqrt{2}[/latex], \\ [latex]\theta = 45^{\circ}[/latex]
  49. [latex]\|\vec{v}\| = \sqrt{17}[/latex], \\ [latex]\theta = \arctan \left( -4 \right) + \pi \approx 284.04^{\circ}[/latex]
  50. [latex]\|\vec{v}\| \approx 145.48[/latex], \\ [latex]\theta = \arctan \left( -\frac{77.05}{123.4} \right) + \pi \approx 328.02^{\circ}[/latex]
  51. [latex]\|\vec{v}\| \approx 1274.00[/latex], \\ [latex]\theta = \arctan \left( \frac{831.6}{965.15} \right) \approx 40.75^{\circ}[/latex]
  52. [latex]\|\vec{v}\| \approx 121.69[/latex], \\ [latex]\theta = \arctan \left(- \frac{42.3}{114.1} \right)\approx 159.66^{\circ}[/latex]
  53. The boat’s true speed is about 10 miles per hour at a heading of S[latex]50.6^{\circ}[/latex]W.
  54. The HMS Sasquatch’s true speed is about 41 miles per hour at a heading of S[latex]26.8^{\circ}[/latex]E.
  55. She should maintain a speed of about 35 miles per hour at a heading of S[latex]11.8^{\circ}[/latex]E.
  56. She should fly at 83.46 miles per hour with a heading of N[latex]22.1^{\circ}[/latex]E
  57. The current is moving at about 10 miles per hour bearing N[latex]54.6^{\circ}[/latex]W.
  58. The tension on each of the cables is about 346 pounds.
  59. The maximum weight that can be held by the cables in that configuration is about 133 pounds.
  60. The tension on the left hand cable is 285.317 lbs. and on the right hand cable is 92.705 lbs.
  61. The weaker student should pull about 60 pounds. The net force on the keg is about 153 pounds.
  62. The resultant force is only about 296 pounds so the couch doesn’t budge. Even if it did move, the stronger force on the third rope would have made the couch drift slightly to the south as it traveled down the street.

Section 9.2 Answers

  1. [latex]\vec{v} = \left\langle -2, -7 \right\rangle[/latex] and [latex]\vec{w} = \left\langle 5, -9 \right\rangle[/latex]
    [latex]\vec{v} \cdot \vec{w} = 53[/latex]
    [latex]\theta = 45^{\circ}[/latex]
    [latex]\text{proj}_{\vec{w}}(\vec{v}) = \left\langle \frac{5}{2}, -\frac{9}{2} \right\rangle[/latex]
    [latex]\vec{q} = \left\langle -\frac{9}{2}, -\frac{5}{2} \right\rangle[/latex]
  2. [latex]\vec{v} = \left\langle -6, -5 \right\rangle[/latex] and [latex]\vec{w} = \left\langle 10, -12 \right\rangle[/latex]
    [latex]\vec{v} \cdot \vec{w} = 0[/latex]
    [latex]\theta = 90^{\circ}[/latex]
    [latex]\text{proj}_{\vec{w}}(\vec{v}) = \left\langle 0, 0 \right\rangle[/latex]
    [latex]\vec{q} = \left\langle -6, -5 \right\rangle[/latex]
  3. [latex]\vec{v} = \left\langle 1, \sqrt{3} \right\rangle[/latex] and [latex]\vec{w} = \left\langle 1, -\sqrt{3} \right\rangle[/latex]
    [latex]\vec{v} \cdot \vec{w} = -2[/latex]
    [latex]\theta = 120^{\circ}[/latex]
    [latex]\text{proj}_{\vec{w}}(\vec{v}) = \left\langle -\frac{1}{2}, \frac{\sqrt{3}}{2} \right\rangle[/latex]
    [latex]\vec{q} = \left\langle \frac{3}{2}, \frac{\sqrt{3}}{2} \right\rangle[/latex]
  4. [latex]\vec{v} = \left\langle 3,4 \right\rangle[/latex] and [latex]\vec{w} = \left\langle -6, -8 \right\rangle[/latex]
    [latex]\vec{v} \cdot \vec{w} = -50[/latex]
    [latex]\theta = 180^{\circ}[/latex]
    [latex]\text{proj}_{\vec{w}}(\vec{v}) = \left\langle 3, 4 \right\rangle[/latex]
    [latex]\vec{q} = \left\langle0, 0\right\rangle[/latex]
  5. [latex]\vec{v} = \left\langle -2,1 \right\rangle[/latex] and [latex]\vec{w} = \left\langle 3,6 \right\rangle[/latex]
    [latex]\vec{v} \cdot \vec{w} = 0[/latex]
    [latex]\theta = 90^{\circ}[/latex]
    [latex]\text{proj}_{\vec{w}}(\vec{v}) = \left\langle 0, 0 \right\rangle[/latex]
    [latex]\vec{q} = \left\langle -2, 1 \right\rangle[/latex]
  6. [latex]\vec{v} = \left\langle -3\sqrt{3}, 3\right\rangle[/latex] and [latex]\vec{w} = \left\langle -\sqrt{3}, -1 \right\rangle[/latex]
    [latex]\vec{v} \cdot \vec{w} = 6[/latex]
    [latex]\theta = 60^{\circ}[/latex]
    [latex]\text{proj}_{\vec{w}}(\vec{v}) = \left\langle -\frac{3\sqrt{3}}{2}, -\frac{3}{2} \right\rangle[/latex]
    [latex]\vec{q} = \left\langle -\frac{3\sqrt{3}}{2}, \frac{9}{2} \right\rangle[/latex]
  7. [latex]\vec{v} = \left\langle 1, 17 \right\rangle[/latex] and [latex]\vec{w} = \left\langle -1, 0 \right\rangle[/latex]
    [latex]\vec{v} \cdot \vec{w} = -1[/latex]
    [latex]\theta \approx 93.37^{\circ}[/latex]
    [latex]\text{proj}_{\vec{w}}(\vec{v}) = \left\langle 1, 0 \right\rangle[/latex]
    [latex]\vec{q} = \left\langle 0, 17 \right\rangle[/latex]
  8. [latex]\vec{v} = \left\langle 3, 4 \right\rangle[/latex] and [latex]\vec{w} = \left\langle 5, 12 \right\rangle[/latex]
    [latex]\vec{v} \cdot \vec{w} = 63[/latex]
    [latex]\theta \approx 14.25^{\circ}[/latex]
    [latex]\text{proj}_{\vec{w}}(\vec{v}) = \left\langle \frac{315}{169}, \frac{756}{169} \right\rangle[/latex]
    [latex]\vec{q} = \left\langle \frac{192}{169}, -\frac{80}{169} \right\rangle[/latex]
  9. [latex]\vec{v} = \left\langle -4, -2 \right\rangle[/latex] and [latex]\vec{w} = \left\langle 1, -5 \right\rangle[/latex]
    [latex]\vec{v} \cdot \vec{w} = 6[/latex]
    [latex]\theta \approx 74.74^{\circ}[/latex]
    [latex]\text{proj}_{\vec{w}}(\vec{v}) = \left\langle \frac{3}{13}, -\frac{15}{13} \right\rangle[/latex]
    [latex]\vec{q} = \left\langle -\frac{55}{13}, -\frac{11}{13} \right\rangle[/latex]
  10. [latex]\vec{v} = \left\langle -5, 6 \right\rangle[/latex] and [latex]\vec{w} = \left\langle 4, -7 \right\rangle[/latex]
    [latex]\vec{v} \cdot \vec{w} = -62[/latex]
    [latex]\theta \approx 169.94^{\circ}[/latex]
    [latex]\text{proj}_{\vec{w}}(\vec{v}) = \left\langle -\frac{248}{65}, \frac{434}{65} \right\rangle[/latex]
    [latex]\vec{q} = \left\langle -\frac{77}{65}, -\frac{44}{65} \right\rangle[/latex]
  11. [latex]\vec{v} = \left\langle -8, 3 \right\rangle[/latex] and [latex]\vec{w} = \left\langle 2, 6 \right\rangle[/latex]
    [latex]\vec{v} \cdot \vec{w} = 2[/latex]
    [latex]\theta \approx 87.88^{\circ}[/latex]
    [latex]\text{proj}_{\vec{w}}(\vec{v}) = \left\langle \frac{1}{10}, \frac{3}{10} \right\rangle[/latex]
    [latex]\vec{q} = \left\langle -\frac{81}{10}, \frac{27}{10} \right\rangle[/latex]
  12. [latex]\vec{v} = \left\langle 34, -91 \right\rangle[/latex] and [latex]\vec{w} = \left\langle 0, 1 \right\rangle[/latex]
    [latex]\vec{v} \cdot \vec{w} = -91[/latex]
    [latex]\theta \approx 159.51^{\circ}[/latex]
    [latex]\text{proj}_{\vec{w}}(\vec{v}) = \left\langle 0, -91 \right\rangle[/latex]
    [latex]\vec{q} = \left\langle 34, 0 \right\rangle[/latex]
  13. [latex]\vec{v} =3 \hat{\textbf{i}}- \hat{\textbf{j}}[/latex] and [latex]\vec{w} = 4 \hat{\textbf{j}}[/latex]
    [latex]\vec{v} \cdot \vec{w} = -4[/latex]
    [latex]\theta \approx 108.43^{\circ}[/latex]
    [latex]\text{proj}_{\vec{w}}(\vec{v}) = \left\langle 0,-1 \right\rangle[/latex]
    [latex]\vec{q} = \left\langle 3,0 \right\rangle[/latex]
  14. [latex]\vec{v} = -24 \hat{\textbf{i}}+ 7 \hat{\textbf{j}}[/latex] and [latex]\vec{w} = 2 \hat{\textbf{i}}[/latex]
    [latex]\vec{v} \cdot \vec{w} = -48[/latex]
    [latex]\theta \approx 163.74^{\circ}[/latex]
    [latex]\text{proj}_{\vec{w}}(\vec{v}) = \left\langle -24,0 \right\rangle[/latex]
    [latex]\vec{q} = \left\langle 0,7\right\rangle[/latex]
  15. [latex]\vec{v} =\frac{3}{2} \hat{\textbf{i}}+ \frac{3}{2} \hat{\textbf{j}}[/latex] and [latex]\vec{w} = \hat{\textbf{i}}- \hat{\textbf{j}}[/latex]
    [latex]\vec{v} \cdot \vec{w} = 0[/latex]
    [latex]\theta = 90^{\circ}[/latex]
    [latex]\text{proj}_{\vec{w}}(\vec{v}) = \left\langle 0,0 \right\rangle[/latex]
    [latex]\vec{q} = \left\langle \frac{3}{2},\frac{3}{2} \right\rangle[/latex]
  16. [latex]\vec{v} = 5 \hat{\textbf{i}}+ 12 \hat{\textbf{j}}[/latex] and [latex]\vec{w} = -3 \hat{\textbf{i}}+ 4 \hat{\textbf{j}}[/latex]
    [latex]\vec{v} \cdot \vec{w} = 33[/latex]
    [latex]\theta \approx 59.49^{\circ}[/latex]
    [latex]\text{proj}_{\vec{w}}(\vec{v}) = \left\langle -\frac{99}{25}, \frac{132}{25} \right\rangle[/latex]
    [latex]\vec{q} = \left\langle \frac{224}{25},\frac{168}{25} \right\rangle[/latex]
  17. [latex]\vec{v} = \left\langle \frac{1}{2}, \frac{\sqrt{3}}{2} \right\rangle[/latex] and [latex]\vec{w} = \left\langle -\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right\rangle[/latex]
    [latex]\vec{v} \cdot \vec{w} = \frac{\sqrt{6} - \sqrt{2}}{4}[/latex]
    [latex]\theta = 75^{\circ}[/latex]
    [latex]\text{proj}_{\vec{w}}(\vec{v}) = \left\langle \frac{1-\sqrt{3}}{4}, \frac{\sqrt{3} - 1}{4} \right\rangle[/latex]
    [latex]\vec{q} = \left\langle \frac{1+\sqrt{3}}{4}, \frac{1 +\sqrt{3}}{4} \right\rangle[/latex]
  18. [latex]\vec{v} = \left\langle \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right\rangle[/latex] and [latex]\vec{w} = \left\langle \frac{1}{2}, -\frac{\sqrt{3}}{2} \right\rangle[/latex]
    [latex]\vec{v} \cdot \vec{w} = \frac{\sqrt{2} - \sqrt{6}}{4}[/latex]
    [latex]\theta = 105^{\circ}[/latex]
    [latex]\text{proj}_{\vec{w}}(\vec{v}) = \left\langle \frac{\sqrt{2}-\sqrt{6}}{8}, \frac{3\sqrt{2} - \sqrt{6}}{8} \right\rangle[/latex]
    [latex]\vec{q} = \left\langle \frac{3\sqrt{2}+\sqrt{6}}{8}, \frac{\sqrt{2} + \sqrt{6}}{8} \right\rangle[/latex]
  19. [latex]\vec{v} = \left\langle \frac{\sqrt{3}}{2}, \frac{1}{2} \right\rangle[/latex] and [latex]\vec{w} = \left\langle -\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \right\rangle[/latex]
    [latex]\vec{v} \cdot \vec{w} = -\frac{\sqrt{6} + \sqrt{2}}{4}[/latex]
    [latex]\theta = 165^{\circ}[/latex]
    [latex]\text{proj}_{\vec{w}}(\vec{v}) = \left\langle \frac{\sqrt{3} + 1}{4}, \frac{\sqrt{3} + 1}{4} \right\rangle[/latex]
    [latex]\vec{q} = \left\langle \frac{\sqrt{3} - 1}{4}, \frac{1 - \sqrt{3}}{4} \right\rangle[/latex]
  20. [latex]\vec{v} = \left\langle \frac{1}{2}, -\frac{\sqrt{3}}{2} \right\rangle[/latex] and [latex]\vec{w} = \left\langle \frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \right\rangle[/latex]
    [latex]\vec{v} \cdot \vec{w} = \frac{\sqrt{6} + \sqrt{2}}{4}[/latex]
    [latex]\theta = 15^{\circ}[/latex]
    [latex]\text{proj}_{\vec{w}}(\vec{v}) = \left\langle \frac{\sqrt{3} + 1}{4}, -\frac{\sqrt{3} + 1}{4} \right\rangle[/latex]
    [latex]\vec{q} = \left\langle \frac{1 - \sqrt{3}}{4}, \frac{1 - \sqrt{3}}{4} \right\rangle[/latex]
  21. [latex](1500 \, \text{pounds})(300 \, \text{feet})\cos\left(0^{\circ}\right) = 450,000[/latex] foot-pounds
  22. [latex](10 \, \text{pounds})(3 \, \text{feet})\cos\left(0^{\circ}\right) = 30[/latex] foot-pounds
  23. [latex](13 \, \text{pounds})(25 \, \text{feet}) \cos\left(15^{\circ}\right) \approx 313.92[/latex] foot-pounds
  24. [latex](100 \, \text{pounds})(42 \, \text{feet}) \cos\left(13^{\circ}\right) \approx 4092.35[/latex] foot-pounds
  25. [latex](200 \, \text{pounds})(10 \, \text{feet}) \cos\left(77.5^{\circ}\right) \approx 432.88[/latex] foot-pounds

 

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