1. True or False The symmetric equations \(\dfrac{x-3}{2}= \dfrac{y-4}{1}=\dfrac{z+2}{-1}\) represent a line parallel to the vector \(3 \mathbf{i}+4\mathbf{j}-2\mathbf{k}\).

2. True or False Skew lines can sometimes lie in the same plane.

3. True or False If two planes are not parallel, they intersect in a line.

4. True or False The two planes \(p_{1}\): \(2x+3y-z=3\) and \(p_{2}\): \(6x+3y-3z=9\) are parallel.

5. True or False The planes \(x+2y+z=3\) and \(x+2y+z=4\) are parallel and are 1 unit apart.

6. Which of the three representations of a line in space do you like best? Give reasons for your choice.

7. Which of the two representations of a plane in space do you like better? Give reasons for your choice.

8. Two distinct lines in space that do not intersect and are not parallel are called ___________.

9. containing the point \((1,2,3)\) and in the direction of \(2\mathbf{i}-\mathbf{j}+\mathbf{k}\).

10. containing the point \((4,-1,6)\) and in the direction of \(\mathbf{i}+\mathbf{j}\).

11. containing the points \(P_{0}=(1,-1,3)\) and \(P_{1}=(4,2,1)\).

12. containing the points \(P_{0}=(-2,3,0)\) and \(P_{1}=(1,-1,2)\).

13. Find parametric equations of a line containing the point \( (-1,5,6)\) and in the direction of the line \(\dfrac{x+1}{5}=\dfrac{y-2}{4}= \dfrac{z-3}{-3}.\)

14. Find parametric equations of a line containing the point \( (1,-2,-3)\) and in the direction of the line \(\dfrac{x+1}{6}=\dfrac{y+2}{2}=\dfrac{z}{-1}.\)

15. Let \(\dfrac{x-4}{2}=\dfrac{y+1}{-1}=\dfrac{z-2}{2} \) be symmetric equations of a line. Find a vector in the direction of the line, and find two points on this line.

16. Let \(x+1=y+3=\dfrac{z+4}{2}\) be symmetric equations of a line. Find a vector in the direction of the line, and find two points on this line.

17. \(P_{0}=(4,2,1)\) and the line is in the direction of the vector \(\mathbf{D=}\) \(2\mathbf{i}+\mathbf{k}\).

18. \(P_{0}=(-1,2,0)\) and the line is in the direction of the vector \(\mathbf{D=}\) \(2\mathbf{k-j}\).

19. \(P_{0}=(0,0,0)\) and the line is perpendicular to each of the lines \(\dfrac{x+1}{2}=\dfrac{y-1}{3}=\dfrac{z}{-2}\) and \(\dfrac{x-3}{3}= \dfrac{y}{1}=\dfrac{z+1}{2}\).

20. \(P_{0}=(0,0,0)\) and the line is perpendicular to each of the lines \(\dfrac{x+2}{4}=\dfrac{y-1}{2}=z+1\) and \(\dfrac{x+5}{5}=\dfrac{y+1}{3} =z\).

21. \(P_{0}=(1,2,-1)\) and the line is perpendicular to the vectors \(\mathbf{u}=2\mathbf{i}+4\mathbf{j}-2\mathbf{k}\) and \(\ \mathbf{v}=-3 \mathbf{i}-2\mathbf{j}+\mathbf{k}\).

22. \(P_{0}=(-1,3,2)\) and the line is perpendicular to the vectors \(\mathbf{u}=-2\mathbf{i}+2\mathbf{j}-3\mathbf{k}\) and \(\mathbf{v}=4\mathbf{ i}+2\mathbf{j}+\mathbf{k}\).

23. \(\begin{array}[t]{ll} l_{1}\hbox{:} & \mathbf{r}_{1}=2\mathbf{i}-2\mathbf{k}+t_{1}\left( -3\, \mathbf{i}+6\,\mathbf{j}+6\,\mathbf{k}\right) \\ l_{2}\hbox{:} & \mathbf{r}_{2}=6\,\mathbf{i}+2\,\mathbf{j}+5\,\mathbf{k} +t_{2}\,\left( -\mathbf{i}+2\,\mathbf{j}+2\,\mathbf{k}\right) \end{array}\)

(a, b) Parallel

24. \(\begin{array}[t]{ll} l_{1}\hbox{:} & \mathbf{r}_{1}=3\,\mathbf{i}+3\,\mathbf{k}+t_{1}\left( 3 \mathbf{i}+6\,\mathbf{j}-2\,\mathbf{k}\right) \\ l_{2}\hbox{:} & \mathbf{r}_{2}=3\,\mathbf{i}+3\,\mathbf{k}+t_{2}\left( -2\, \mathbf{i}+4\,\mathbf{j}+7\,\mathbf{k}\right) \end{array}\)

25. \(\begin{array}[t]{ll} l_{1}\hbox{:} & \mathbf{r}_{1}=4\mathbf{i}+3\,\mathbf{j}+t_{1}\left( \mathbf{i}-\mathbf{j}+6\mathbf{k}\right) \\ l_{2}\hbox{:} & \mathbf{r}_{2}=4\,\mathbf{i}+3\,\mathbf{j}+2\,\mathbf{k} +t_{2}\left( \mathbf{i}-\mathbf{j}-2\,\mathbf{k}\right) \end{array}\)

(a, b) Intersect when \(t=\dfrac14\) at the point \((17/4,11/4,3/2)\)

26. \(\begin{array}[t]{ll} l_{1}\hbox{:} & \mathbf{r}_{1}=2\mathbf{i}+7\,\mathbf{j}+t_{1}\left( -2\, \mathbf{i}+8\,\mathbf{j}-6\,\mathbf{k}\right) \\ l_{2}\hbox{:} & \mathbf{r}_{2}=6\,\mathbf{i}-5\,\mathbf{j}+t_{2}\,\left( \mathbf{i}-4\,\mathbf{j}+3\mathbf{k}\right) \end{array}\)

27. \(\begin{array}[t]{ll} l_{1}\hbox{:} & \dfrac{x-3}{2}=\dfrac{y+2}{3}=\dfrac{z-1}{4} \\ l_{2}\hbox{:} & \dfrac{x+4}{-4}=\dfrac{y-3}{-6}=\dfrac{z+4}{-8} \end{array}\)

(a, b) Parallel

28. \(\begin{array}[t]{ll} l_{1}\hbox{:} & \dfrac{x}{3}=\dfrac{y-2}{4}=\dfrac{z+4}{1} \\ l_{2}\hbox{:} & \dfrac{x+6}{3}=\dfrac{y+2}{4}=\dfrac{z-3}{2} \end{array}\)

29. \(\begin{array}[t]{ll} l_{1}\hbox{:} & \dfrac{x+1}{5}=\dfrac{y-2}{4}=\dfrac{z-3}{-3} \\ l_{2}\hbox{:} & \dfrac{x+1}{6}=\dfrac{y-2}{3}=\dfrac{z+3}{2} \end{array}\)

(a, b) Skew

30. \(\begin{array}[t]{ll} l_{1}\hbox{:} & \dfrac{x+5}{6}=\dfrac{y-2}{3}=\dfrac{z-4}{-1} \\ l_{2}\hbox{:} & {x}=\dfrac{y-2}{3}=\dfrac{z-8}{2} \end{array}\)

31. Containing the point \((1,-1,2)\) and normal to the vector \(2\mathbf{i}-\mathbf{j}+\mathbf{k}\)

32. Containing the point \((-3,2,1)\) and normal to the vector \( \mathbf{i}+\mathbf{j}-2\mathbf{k}\)

33. Containing the point \((0,5,-2)\) and parallel to the plane \( x+2y-z=6\)

34. Containing the point \((1,-2,0)\) and parallel to the plane \( 2x-y+3z=10\)

35. Containing the point \((2,3,-1)\) and normal to the line \[ \dfrac{x-1}{2}=\dfrac{y-3}{5}=\dfrac{z+1}{-2} \]

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36. Containing the point \((-1,2,3)\) and normal to the line \[ \dfrac{x+5}{3}=\dfrac{y+2}{4}=\dfrac{z-4}{4} \]

37. Parallel to the \(xy\)-plane and containing the point \(\left( 0,0,4\right)\)

38. Parallel to the \(yz\)-plane and containing the point \(\left( 2,0,0\right)\)

39. Parallel to the \(xz\)-plane and containing the point \((1,-2,3)\)

40. Parallel to the \(xy\)-plane and containing the point \((1,-3,4)\)

41. \(P_{1}=(0,0,0)\); \(P_{2}=(1,2,-1)\); \(P_{3}=(-1,1,0)\)

42. \(P_{1}=(0,0,0)\); \(P_{2}=(3,-1,2)\); \(P_{3}=(-3,1,0)\)

43. \(P_{1}=(1,2,1)\); \(P_{2}=(3,2,2)\); \(P_{3}=(4,-1,-1)\)

44. \(P_{1}=(-1,2,0)\); \(P_{2}=(3,4,-1)\); \(P_{3}=(-2,-1,0)\)

45. \(P_{1}=(6,8,-2)\); \(P_{2}=(4,-1,0)\); \(P_{3}=(1,0,0)\)

46. \(P_{1}=(-3,-4,0)\); \(P_{2}=(6,-7,2)\); \(P_{3}=(0,0,1)\)

47. \(p_{1}\): \(x+y-z=-5\) and \(p_{2}\): \(\ 2x+3y-4z=-1\)

48. \(p_{1}\): \(\ 2x-y+z=2\) and \(p_{2}\): \(x+y+z=3\)

49. \(p_{1}\): \(x-y=2\) and \(p_{2}\): \(\ y-z=2\)

50. \(p_{1}\): \(\ 2x-3y+z=1\) and \(p_{2}\): \(\ 2x-3y+4z=2\)

51. from \((1, 2,-1)\) to \(2x-y+z=1\)

52. from \((-1,3,-2)\) to \(x+2y-3z=4\)

53. from \(\left( 2,-1,\,1\right)\) to \(-x+y-3z=6\)

54. from \((-2,1,1)\) to \(-3x+2y+z=1\)

55. \(p_{1}\): \(2x+y-2z=-1\) and \(p_{2}\): \(2x+y-2z=3\)

56. \(p_{1}\): \(x+2y-2z=-3\) and \(p_{2}\): \(\ x+2y-2z=3\)

57. \(p_{1}\): \(x-2z=-1\) and \(p_{2}\): \(x-2z=3\)

58. \(p_{1}\): \(-x+y+2z=-4\) and \(p_{2}\): \(-x+y+2z=-1\)

59. Plane \(2x+y-z=5\), line \(\dfrac{x-1}{2}=\dfrac{y+3}{4}=\dfrac{z-1 }{1}\)

60. Plane \(x+y-2z=8\), line \(\dfrac{x+1}{2}=\dfrac{y-3}{1}=\dfrac{z-4 }{-2}\)

61. Plane \(2x+3y+z=5,\) line \(\dfrac{x-3}{1}=\dfrac{y+4}{2}=\dfrac{ z-1}{2}\)

62. Plane \(x+y-z=3;\) line \(\dfrac{x+2}{2}=\dfrac{y-3}{1}=\dfrac{z}{2 }\)

63. \(\begin{array}[t]{ll} l_{1}\hbox{:} & \mathbf{r}_{1}=(2-t_{1})\mathbf{i}+(4+2t_{1})\mathbf{j} +(-5+t_{1})\mathbf{k} \\ l_{2}\hbox{:} & \mathbf{r}_{2}=(4-t_{2})\mathbf{i}+(3+t_{2})\mathbf{j} +(-13+3t_{2})\mathbf{k} \end{array}\)

64. \(\begin{array}[t]{ll} l_{1}\hbox{:} & \mathbf{r}_{1}=(5+2t_{1})\mathbf{i}+(6-t_{1})\mathbf{j} +2t_{1}\mathbf{k} \\ l_{2}\hbox{:} & \mathbf{r}_{2}=(7+3t_{2})\mathbf{i}+(5-2t_{2})\mathbf{j} +(2-t_{2})\mathbf{k} \end{array}\)

65. \(\begin{array}[t]{ll} l_{1}\hbox{:} & \mathbf{r}_{1}=t_{1}\mathbf{i}+(1+2t_{1})\mathbf{j} +(-3+t_{1})\mathbf{k} \\ l_{2}\hbox{:} & \mathbf{r}_{2}=(-3+t_{2})\mathbf{i}+(1-4t_{2})\mathbf{j} +(2-7t_{2})\mathbf{k} \end{array}\)

66. \(\begin{array}[t]{ll} l_{1}\hbox{:} & \mathbf{r}_{1}=(2-3t_{1})\mathbf{i}+6t_{1}\mathbf{j} +(-2+5t_{1})\mathbf{k} \\ l_{2}\hbox{:} & \mathbf{r}_{2}=-2t_{2}\mathbf{i}+(1+t_{2})\mathbf{j}+2t_{2} \mathbf{k} \end{array}\)

67. 1. (a) Find parametric equations of the line containing the point \((1,2,-1)\) and normal to the plane \(2x-y+z-6=0.\)
    2. (b) Graph the line and the plane.

68. 1. (a) Find parametric equations of the line containing the point \((2,3,-1)\) and normal to the plane \(x+y-z-3=0\).
    2. (b) Graph the line and the plane.

69. What property can you assign to the lines \(l_{1}\) and \(l_{2}\) , given below, if \(a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}=0\)? \[ \begin{array}{ll} l_{1}\hbox{:} & \dfrac{x-x_{1}}{a_{1}}=\dfrac{y-y_{1}}{a_{2}}=\dfrac{ z-z_{1}}{a_{3}} \\ l_{2}\hbox{:} & \dfrac{x-x_{1}}{b_{1}}=\dfrac{y-y_{1}}{b_{2}}=\dfrac{ z-z_{1}}{b_{3}} \end{array} \]

70. What property can you assign to the distinct lines \(l_{1}\) and \( l_{2}\), given below, if \(\dfrac{a_{1}}{b_{1}}=\dfrac{a_{2}}{b_{2}}=\dfrac{ a_{3}}{b_{3}}?\) \[ \begin{array}{ll} l_{1}\hbox{:} & \dfrac{x-x_{1}}{a_{1}}=\dfrac{y-y_{1}}{a_{2}}=\dfrac{z-z_{1} }{a_{3}} \\ l_{2}\hbox{:} & \dfrac{x-x_{2}}{b_{1}}=\dfrac{y-y_{2}}{b_{2}}=\dfrac{ z-z_{2}}{b_{3}} \end{array} \]

71. \(p_{1}\): \(\ 2x-y+z=2\) and \(p_{2}\): \(x+y+z=3\)

72. \(p_{1}\): \(\ x+y-z=5\); and \(p_{2}\): \(\ 2x+3y-4z=1\)

73. \(p_{1}\): \(\ 2x-3y+z=1\) and \(p_{2}\): \(\ 2x-3y+4z=2\)

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74. \(p_{1}\): \(x-y=2\) and \(p_{2}\): \(\ y-z=2\)

75. \(p_{1}\): \(\ 2x-y+z=3\) and \(p_{2}\): \(\ 4x-y+6z=7\)

76. \(p_{1}\): \(\ x+y-z=1\) and \(\ p_{2}\): \(\ 2x-2y+2z=-2\)

77. Paths of Spacecrafts Two unidentified flying objects are at the points \((t,-t, 1-t)\) and \((t-3, 2t, 4t-1)\) at time \(t\), \(t\geq 0.\)

    1. (a) Describe the paths of the objects.
    2. (b) Find the acute angle between the paths.
    3. (c) Find where the paths intersect (or determine that they do not).
    4. (d) Will the objects collide?

78. Find symmetric equations of the line passing through the centers of the spheres \[ x^{2}+y^{2}+z^{2}-2x-4y+4z=8 \]

    and \[ x^{2}+y^{2}+z^{2}+2x+6y+4z=20 \]

79. Find parametric equation of the line perpendicular to the lines \[ l_{1}\hbox{:}\quad x=1-t,\quad y=t,\quad z=2t-1 \]

    and \[ l_{2}:\quad x=t+1,\quad y=-t,\quad z=t-1 \]

    at their point of intersection. Why is this line parallel to the \(xy\)-plane?

80. Explain why the set of points \((x,y,z)\) equidistant from the points \((1,3,0)\) and \((-1,1,2)\) is a plane. Then find its equation in two ways, as follows:

    1. (a) Use the distance formula to equate the distances between \( (x,y,z)\) and the given points, simplifying the result to obtain an equation of the plane.
    2. (b) Find a point on the plane and a vector normal to the plane, and use the answer to find an equation of the plane.

81. Find symmetric equations of the line of intersection of the planes \(2x+y-z=6\) and \(x-y+3z=4\).

82. Find symmetric equations of the line that contains \((2,0,-3)\) , is perpendicular to \(\mathbf{i}+2\mathbf{j}-\mathbf{k}\), and is parallel to the plane \(2x+3y-z=1\).

83. Find the point of intersection of the line through the points \((0,2,-2)\), and \((2,1,-3)\), and the plane through the points \((0,4,-2)\), \((1,3,-2)\), and \( (2,2,-3)\).

84. Find an equation of the plane parallel to the line \(\mathbf{r} =2\mathbf{i}+t(-\mathbf{i}+\mathbf{j}+2\mathbf{k})\) and containing the points \((2,2,-1)\) and \((1,0,1)\).

85. Find an equation of the plane tangent to the sphere \( (x-1)^{2}+(y+2)^{2}+(z-2)^{2}=6\) at the point \((2,-1, 0)\).

86. Sphere Find an equation of the sphere with its center at \((-2,1,5)\) and tangent to \(x+2y-2z=8\).

87. Find symmetric equations of a line normal to the plane containing the lines \[ x-2=\frac{y+1}{2}=\frac{z-1}{2}\qquad \hbox{and }\qquad x+1=\frac{y+8}{3}= \frac{z}{-3} \]

    at their point of intersection.

88. Find an equation for the plane containing the origin that is perpendicular to the plane \(x-2y-z=0\) and makes an angle of \(60^{\circ}\) with the positive \(y\)-axis.

89. We proved that the distance between the point \( P_{0}=(x_{0}, y_{0}, z_{0})\) and the plane \(Ax+By+Cz=D\) is \[ d=\frac{|Ax_{0}+By_{0}+Cz_{0}-D|}{\sqrt{A^{2}+B^{2}+C^{2}}} \]

    Derive this formula differently, as follows:

    1. (a) Show that the line through \(P_{0}\) normal to the plane has the parametric equations \(x=x_{0}+At\), \(y=y_{0}+Bt\), \(z=z_{0}+Ct\).
    2. (b) If \(P=(x, y, z)\) is the point of intersection of the line in (a) with the plane, show that \(x-x_{0}=At\), \(y-y_{0}=Bt\), \(z-z_{0}=Ct\), where \[ t=\frac{-(Ax_{0}+By_{0}+Cz_{0})+D}{A^{2}+B^{2}+C^{2}} \]
    3. (c) Explain why \(d\) is the distance between \(P_{0}\) and \(P\), and use the distance formula to finish the proof.

90. Prove that the distance between the parallel planes \( Ax+By+Cz=D_{1}\) and \(Ax+By+Cz=D_{2}\) is \[ d=\frac{|D_{2}-D_{1}|}{\sqrt{A^{2}+B^{2}+C^{2}}} \]

91. Let two skew lines have respective direction vectors \(\mathbf{ D}_{1}\) and \(\mathbf{D}_{2}\). Let \(A\) and \(B\) be points on the respective lines, and let \(\mathbf{w}=\skew5\overrightarrow{\it AB}\). Show that the distance between the two lines is the magnitude of the vector projection of \(\mathbf{w }\) onto \(\mathbf{D}_{1}\times \mathbf{D}_{2}\): \[ |\hbox{proj}_{\mathbf{D}_{1}\times \mathbf{D}_{2}}\mathbf{w}|=\frac{|\mathbf{ w}\,{\cdot}\, (\mathbf{D}_{1}\times \mathbf{D}_{2})|}{\Vert \mathbf{D}_{1}\times \mathbf{D}_{2}\Vert } \]

    (The shortest distance is measured along the common perpendicular to the two lines.)

92. Find the distance between the lines \(\dfrac{x-3}{2}=\dfrac{y }{3}=z\qquad\) and \(\qquad x=\dfrac{y+1}{-2}=\dfrac{z-2}{-1}\).

93. What is the minimum distance between the skew lines \( x-1=y-2=z+6\) and \(\dfrac{x-1}{2}=\dfrac{y+2}{-3}=z-10\)? Locate the points on each line at which the distance is minimum.