11.5.2: Homework

Reading Questions

1. What are the two vector components of a two-dimensional vector \(\vec{v}\)?
2. If a vector \(\vec{v}\) has magnitude \(||\vec{v}||\) and direction angle \(\theta\), what are the formulas for its scalar components \(v_x\) and \(v_y\)?
3. What is a unit vector?
4. What is the process of "normalizing" a vector?
5. How do you write a vector \(\vec{v}\) in coordinate form?
6. If you have a vector in coordinate form, \(\vec{v} = v_x\vec{i} + v_y\vec{j}\), how do you find its magnitude?
7. How do you add two vectors when they are in coordinate form?
8. How do you perform scalar multiplication on a vector that is in coordinate form?
9. What does it mean for an object to be in static equilibrium?
10. What is the zero vector?

Skills Refresher

Review the following skills you will need for this section.

Skills Refresher

1. State the sum of angles formula for sine.

2. State the sum of angles formula for cosine.

3. State the sum of angles formula for tangent.

4. Explain how the difference of angles formulas differ from the sum of angles formulas.

For the following exercises, let \(\cos \left(\alpha\right)= -\dfrac{2}{\sqrt{5}}\), where \(90^{\circ}<\alpha<180^{\circ}\).

5. Find an exact value for \(\cos \left(\alpha+30^{\circ}\right)\).

6. Find an exact value for \(\tan \left(\alpha-30^{\circ}\right)\).

Answers

1. \(\sin (\alpha+\beta)=\sin \left(\alpha\right) \cos \left(\beta\right)+\cos \left(\alpha\right) \sin \left(\beta\right)\)

2. \(\cos (\alpha+\beta)=\cos \left(\alpha\right) \cos \left(\beta\right)-\sin \left(\alpha\right) \sin \left(\beta\right)\)

3. \(\tan (\alpha+\beta)=\dfrac{\tan \left(\alpha\right)+\tan \left(\beta\right)}{1 \tan \left(\alpha\right) \tan \left(\beta\right)}\)

4. Change subtraction signs to addition and vice versa.

5. \(\dfrac{-2 \sqrt{3}-1}{2 \sqrt{5}}\)

6. \(\dfrac{-\sqrt{3}-2}{1+2 \sqrt{3}}\)

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Homework

Vocabulary Check

1. A vector having a magnitude of 1 is called a ___ vector.

2. The ___ form of a vector expresses the vector in terms of \( \vec{i} \) and \( \vec{j} \).

3. Then ___ vector has magnitude 0.

4. \( \vec{v}_x \) and \( \vec{v}_y \) are called the ___ of the vector \( \vec{v} \), while \( v_x \) and \( v_y \) are simply called the ___.

Concept Check

5. Is it easier to add vectors in geometric form or coordinate form? Why?

6. To find a unit vector in the direction of \(\vec{v}\), multiply the coordinates of \(\vec{v}\) by ___ .

7. To find a vector of length \(k\) in the direction of \(\vec{v}\), multiply the coordinates of \(\vec{v}\) by ___.

8. Name two physical quantities that are represented by vectors.

9. What are the components of a vector?

10. Does \(\norm{\vec{v}} = \norm{ \vec{v}_x} + \norm{\vec{v}_y} \)?

True or False? For the following exercises, determine if the statement is true or false. If true, cite the definition or theorem stated in the text supporting your claim. If false, explain why it is false and, if possible, correct the statement.

11. Any vector can be written as the sum of its horizontal and vertical vector components, \(\vec{v}_{x}\) and \(\vec{v}_{y}\).

12. The component vectors of a vector \(\vec{v}\) whose direction is given by the angle \(\theta\) in standard position are the scalar quantities\[\begin{array}{rcl}
    \vec{v}_x & = & \norm{\vec{v}} \sin\left( \theta \right) \\[6pt] \vec{v}_y & = & \norm{\vec{v}} \cos\left( \theta \right) \\[6pt] \end{array} \nonumber \]

13. The magnitude and direction of a vector with components \( \norm{\vec{v}_x} \) and \( \norm{\vec{v}_x} \) are given by\[ \norm{\vec{v}} = \sqrt{\left(\norm{\vec{v}_x}\right)^2 + \left(\norm{\vec{v}_y}\right)^2} \quad \text {and} \quad \tan\left( \theta \right)=\dfrac{\norm{\vec{v}_y}}{\norm{\vec{v}_x}}, \nonumber \]respectively.

Basic Skills

For the following exercises, use the coordinate form of each vector shown in the figure to answer the question.

14.   

    1. \(\norm{\vec{u}}\)

    2. \(2 \vec{u}\)

    3. \(\norm{2 \vec{u}}\)

15.    

    1. \(\norm{\vec{v}}\)

    2. \(\frac{1}{2} \vec{v}\)

    3. \(\norm{\frac{1}{2} \vec{v}}\)

16.    

    1. \(\norm{\vec{w}}\)

    2. \(-\vec{w}\)

    3. \(\norm{-\vec{w}}\)

17.     

    1. \(\norm{\vec{z}}\)

    2. \(-3 z\)

    3. \(\norm{-3\vec{z}}\)

18.    

    1. Calculate \(\vec{u}+\vec{v}\) and \(\norm{\vec{u}+\vec{v}}\).

    2. Which of the following statements is true?

       1. \(\norm{\vec{u}}+\norm{\vec{v}} \leq\norm{\vec{u}+\vec{v}} \)

       2. \(\norm{\vec{u}}+\norm{\vec{v}} =\norm{\vec{u}+\vec{v}} \)

       3. \(\norm{\vec{u}}+\norm{\vec{v}} \geq \norm{\vec{u}+\vec{v}}\)

19.    

    1. Calculate \(\vec{w}+\vec{z}\) and \(\norm{\vec{w}+\vec{z}}\).

    2. Which of the following statements is true?

       1. \(\norm{\vec{w}}+\norm{\vec{z}} \leq \norm{\vec{w}+\vec{z}}\)

       2. \(\norm{\vec{w}}+\norm{\vec{z}} =\norm{\vec{w}+\vec{z}}\)

       3. \(\norm{\vec{w}}+\norm{\vec{z}} \geq\norm{\vec{w}+\vec{z}}\)

For the following exercises,
(a) Sketch the vector and give its coordinate form.
(b) Find the magnitude and direction of the vector.

20. The displacement vector from (1, −2) to (−4, 6).

21. The displacement vector from (−5, 2) to (4, 7).

22. The displacement vector from (−2, 9) to (−4, 8).

23. The displacement vector from (−6, 2) to (3, 0).

24. Camille is 12 meters east and 3 meters north of Roy. Lap is 6 meters east and 9 meters north of Camille.

    1. Calculate the displacement vector from Roy to Lap in coordinate form. Let \(\vec{i}\) point east and \(\vec{j}\) point north.

    2. Find the magnitude and direction of the displacement vector.

25. Ron and Sang are climbing a rock wall. Ron is 8 feet to the right and and 23 feet above their starting point. Sang is 2 feet to the right and 7 feet above Ron.

    1. Calculate the displacement vector from the starting point to Sang in coordinate form. Let \(\vec{i}\) point right and \(\vec{j}\) point up.

    2. Find the magnitude and direction of the displacement vector.

For the following exercises, find the magnitude and direction of the vector.

26. \(\vec{v}=-6 \vec{i}+6 \vec{j}\)

27. \(\vec{p}=-12 \vec{i}-5 \vec{j}\)

28. \(\vec{w}=7 \sqrt{3} \vec{i}-7 \vec{j}\)

29. \(z=-6 \sqrt{2} \vec{i}+6 \sqrt{6} \vec{j}\)

30. \(\vec{q}=52 \vec{i}+96 \vec{j}\)

31. \(\vec{s}=3.2 \vec{i}-1.8 \vec{j}\)

For the following exercises, find the coordinate form of the vector.

32. \(\norm{\vec{v}}=6, \quad \theta=-45^{\circ}\)

33. \(\norm{\vec{v}}=200, \quad \theta=240^{\circ}\)

34. \(\norm{\vec{v}}=8.3, \quad \theta=37^{\circ}\)

35. \(\norm{\vec{v}}=23, \quad \theta=200^{\circ}\)

For the following exercises, sketch each vector and its components. Use the coordinate form to find the resultant vector \(\vec{u}+\vec{v}\), and sketch it.

36. \(\vec{u}=-3 \vec{i}+2 \vec{j}, \quad \vec{v}=4 \vec{i}-4 \vec{j}\)

37. \(\vec{u}=5 \vec{i}+\vec{j}, \quad \vec{v}=2 \vec{i}-3 \vec{j}\)

38. \(\vec{u}=-5 \vec{i}-2 \vec{j}, \quad \vec{v}=\vec{i}+6 \vec{j}\)

39. \(\vec{u}=8 \vec{i}-3 \vec{j}, \quad \vec{v}=-4 \vec{i}-2 \vec{j}\)

For the following exercises, find the sum \(\vec{u}+\vec{v}\) of the given vectors.

40. \(\vec{u}=13 \vec{i}-8 \vec{j}, \quad \vec{v}=-1 \vec{i}+11 \vec{j}\)

41. \(\vec{u}=3.7 \vec{i}+2.6 \vec{j}, \quad \vec{v}=-1.3 \vec{i}-5.7 \vec{j}\)

42. \(\vec{u}=-3 \vec{i}+9 \vec{j}, \quad \vec{v}=5.8 \vec{i}-7.1 \vec{j}\)

43. \(\vec{u}=6 \vec{i}-8 \vec{j}, \quad \vec{v}=23 \vec{i}+42 \vec{j}\)

For the following exercises, find the coordinate form of the vector, where\[\vec{u}=2 \vec{i}+3 \vec{j}, \quad \vec{v}=-5 \vec{i}+4 \vec{j}, \quad \vec{w}=-2 \vec{i}-5 \vec{j}, \quad \vec{z}=8 \vec{i}-3 \vec{j}\nonumber \]

44. \(\vec{u}+\vec{v}\)

45. \(\vec{w}-\vec{z}\)

46. \(4 \vec{w}\)

47. \(-3 \vec{v}\)

48. \(2 \vec{z}-\vec{u}\)

49. \(-\vec{w}+5 \vec{u}\)

50. \(3 \vec{v}-\vec{w}+2 \vec{u}\)

51. \(\vec{z}-2(\vec{v}+\vec{w})\)

For the following exercises, find a unit vector \(\vec{u}\) in the same direction as the given vector.

52. \(\vec{r}=-12 \vec{i}+5 \vec{j}\)

53. \(\vec{s}=7 \vec{i}-24 \vec{j}\)

54. \(\vec{t}=\vec{i}-\vec{j}\)

55. \(\vec{w}=-2 \vec{i}-3 \vec{j}\)

For the following exercises, find a vector \(\vec{v}\) in the same direction as \(\vec{w}\), but with the given length.

56. \(\vec{w}=8 \vec{i}+15 \vec{j},\quad \norm{\vec{v}}=51\)

57. \(\vec{w}=-20 \vec{i}-21 \vec{j}, \quad \norm{\vec{v}}=58\)

58. \(\vec{w}=-3 \vec{i}+\vec{j}, \quad \norm{\vec{v}}=4\)

59. \(\vec{w}=\vec{i}-2 \vec{j}, \quad \norm{\vec{v}}=7\)

For the following exercises,
(a) Draw a diagram using arrows to represent the vectors.
(b) Convert each vector to coordinate form.
(c) Use the coordinate form to add or subtract the vectors.

60. Find \(\vec{u}+\vec{v}\), where \(\vec{u}\) has magnitude 2.6 and direction \(\theta=23^{\circ}, \vec{v}\) has magnitude 5.8 and direction \(\theta=223^{\circ}\).

61. Find \(\vec{u}+\vec{v}\), where \(\vec{u}\) has magnitude 50 and direction \(\theta=173^{\circ}, \vec{v}\) has magnitude 70 and direction \(\theta=308^{\circ}\).

62. Find \(\vec{u}-\vec{v}\), where \(\vec{u}\) has magnitude 35 and direction \(\theta=110^{\circ}, \vec{v}\) has magnitude 60 and direction \(\theta=165^{\circ}\).

63. Find \(\vec{u}-\vec{v}\), where \(\vec{u}\) has magnitude 12.4 and direction \(\theta=250^{\circ}, \vec{v}\) has magnitude 8.8 and direction \(\theta=315^{\circ}\).

For the following exercises,
(a) Find the resultant force.
(b) Find the additional force needed for the system to be in equilibrium.

64. \(\vec{F}_1 = -3\vec{i} + \vec{j}, \quad \vec{F}_2 = 5\vec{i}-2\vec{j}, \quad \vec{F}_3 = -6\vec{i} - 4\vec{j}\)

65. \(\vec{F}_1 = 10\vec{i} + 4\vec{j}, \quad \vec{F}_2 = -12\vec{i} - 9\vec{j}, \quad \vec{F}_3 = -3\vec{i} + 5\vec{j}\)

66.  

    

67.  

    

68.    

    1. Find the magnitude of the vector \(\vec{v} =6 \vec{i}-8 \vec{j}\), and the magnitude of the vector \(2 \vec{v}=12 \vec{i}-16 \vec{j}\), and verify that \(\norm{2 \vec{v}}=2\norm{\vec{v}}\).

    2. Let \(\vec{v}=a \vec{i}+b \vec{j}\), and verify that \(\norm{k \vec{v}}=k\norm{\vec{v}}\), for \(k>0\).

69.    

    1. Find the magnitude of the vector \(\vec{v} =3 \vec{i}+5 \vec{j}\), and verify that the vector \(\frac{\vec{v}}{\norm{\vec{v}}}\) has magnitude 1.

    2. Let \(\vec{v}=a \vec{i}+b \vec{j}\), where \(a\) and \(b\) are not both 0, and verify that \(\frac{\vec{v}}{\norm{\vec{v}}}\) is a unit vector.

Synthesis Questions

70. Find the horizontal and vertical components of \(\vec{u}, \vec{v}\), and \(\vec{A}\) from Problem 55 . What do you notice when you compare the horizontal components of two vectors with the horizontal component of the difference?

71. Find the horizontal and vertical components of \(\vec{z}, \vec{y}\), and \(\vec{B}\) from Problem 56 . What do you notice when you compare the horizontal components of two vectors with the horizontal component of the difference?

Applications

For the following exercises,
(a) Make a sketch using vectors to illustrate the problem.
(b) Use the coordinate form of the vectors to solve problem.

72. The tornado displaced the trash bin to a spot 500 meters north and 800 meters east of its original position, and the flood later displaced the bin 2000 meters due south from there. How far and in what direction was the trash bin moved from it original position?

73. A radio-controlled model plane pointed due west with an airspeed of 15 miles per hour, but there was a crosswind from the north at a speed of 8 miles per hour. How fast and in what direction is the plane moving relative to the ground?

74. Matt flies \(10\) km in a direction \(175^{\circ}\) north from east, then turns and flies an additional \(12\) km due west. How far and in what direction is Matt's final position relative to his starting point?

75. Kim sails 500 yards due south, then turns and sails 350 yards in the direction \(300^{\circ}\) from east. How far and in what direction is Kim's final position relative to her starting point?

76. After leaving the airport, Piyali flew 30 miles at a heading \(30^{\circ}\) east of north, then 50 miles \(70^{\circ}\) east of north, and finally 12 miles \(20^{\circ}\) south of east. What is her current position relative to the airport?

77. On a whale-watching trip, the SS Dolphin sailed 15 miles from port on a heading of \(40^{\circ}\), then 8 miles on a heading of \(320^{\circ}\), and then 4 miles on a heading of \(250^{\circ}\). What is the current position of the SS Dolphin relative to port?