Stewart Section 12.5: Equations of Lines and Planes

83문제

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Stewart Section 12.5: Equations of Lines and Planes 0/83
1 Lines and Planes - True/False · Level 2
Determine whether each statement is true or false in \(RR^3\).
(a) Two lines parallel to a third line are parallel.
(b) Two lines perpendicular to a third line are parallel.
(c) Two planes parallel to a third plane are parallel.
(d) Two planes perpendicular to a third plane are parallel.
(e) Two lines parallel to a plane are parallel.
(f) Two lines perpendicular to a plane are parallel. (g) Two planes parallel to a line are parallel. (h) Two planes perpendicular to a line are parallel. (i) Two planes either intersect or are parallel. (j) Two lines either intersect or are parallel. (k) A plane and a line either intersect or are parallel.

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2 Lines and Planes - Vector Equations · Level 2
Find a vector equation and parametric equations for the line through the point \((6, -5, 2)\) and parallel to the vector \(\langle 1, 3, -\dfrac{2}{3} \rangle\).
3 Lines and Planes - Vector Equations · Level 2
Find a vector equation and parametric equations for the line through the point \((2, 2.4, 3.5)\) and parallel to the vector \(3 \mathbf{i} + 2 \mathbf{j} - \mathbf{k}\).
4 Lines and Planes - Vector Equations · Level 2
Find a vector equation and parametric equations for the line through the point \((0, 14, -10)\) and parallel to the line \(x = -1 + 2t\), \(y = 6 - 3t\), \(z = 3 + 9t\).
5 Lines and Planes - Vector Equations · Level 3
Find a vector equation and parametric equations for the line through the point \((1, 0, 6)\) and perpendicular to the plane \(x + 3y + z = 5\).
6 Lines and Planes - Parametric and Symmetric Equations · Level 2
Find parametric equations and symmetric equations for the line through the origin and the point \((4, 3, -1)\).
7 Lines and Planes - Parametric and Symmetric Equations · Level 2
Find parametric equations and symmetric equations for the line through the points \(\left(0, \dfrac{1}{2}, 1\right)\) and \((2, 1, -3)\).
8 Lines and Planes - Parametric and Symmetric Equations · Level 2
Find parametric equations and symmetric equations for the line through the points \((1, 2.4, 4.6)\) and \((2.6, 1.2, 0.3)\).
9 Lines and Planes - Parametric and Symmetric Equations · Level 2
Find parametric equations and symmetric equations for the line through the points \((-8, 1, 4)\) and \((3, -2, 4)\).
10 Lines and Planes - Parametric and Symmetric Equations · Level 3
Find parametric equations and symmetric equations for the line through \((2, 1, 0)\) and perpendicular to both \(\mathbf{i} + \mathbf{j}\) and \(\mathbf{j} + \mathbf{k}\).
11 Lines and Planes - Parametric and Symmetric Equations · Level 3
Find parametric equations and symmetric equations for the line through \((-6, 2, 3)\) and parallel to the line \(\dfrac{1}{2} x = \dfrac{1}{3} y = z + 1\).
12 Lines and Planes - Parametric and Symmetric Equations · Level 3
Find parametric equations and symmetric equations for the line of intersection of the planes \(x + 2y + 3z = 1\) and \(x - y + z = 1\).
13 Lines and Planes - Line Properties · Level 3
Is the line through \((-4, -6, 1)\) and \((-2, 0, -3)\) parallel to the line through \((10, 18, 4)\) and \((5, 3, 14)\)?
14 Lines and Planes - Line Properties · Level 3
Is the line through \((-2, 4, 0)\) and \((1, 1, 1)\) perpendicular to the line through \((2, 3, 4)\) and \((3, -1, -8)\)?
15 Lines and Planes - Line Properties · Level 3
(a) Find symmetric equations for the line that passes through the point \((1, -5, 6)\) and is parallel to the vector \(\langle -1, 2, -3 \rangle\).
(b) Find the points in which the required line in part (a) intersects the coordinate planes.

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16 Lines and Planes - Line Properties · Level 3
(a) Find parametric equations for the line through \((2, 4, 6)\) that is perpendicular to the plane \(x - y + 3z = 7\).
(b) In what points does this line intersect the coordinate planes?

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17 Lines and Planes - Line Segments · Level 2
Find a vector equation for the line segment from \((6, -1, 9)\) to \((7, 6, 0)\).
18 Lines and Planes - Line Segments · Level 2
Find parametric equations for the line segment from \((-2, 18, 31)\) to \((11, -4, 48)\).
19 Lines and Planes - Line Classification · Level 3
Determine whether the lines \(L_1\) and \(L_2\) are parallel, skew, or intersecting. If they intersect, find the point of intersection. \(L_1\): \(x = 3 + 2t\), \(y = 4 - t\), \(z = 1 + 3t\) \(L_2\): \(x = 1 + 4s\), \(y = 3 - 2s\), \(z = 4 + 5s\)
20 Lines and Planes - Line Classification · Level 3
Determine whether the lines \(L_1\) and \(L_2\) are parallel, skew, or intersecting. If they intersect, find the point of intersection. \(L_1\): \(x = 5 - 12t\), \(y = 3 + 9t\), \(z = 1 - 3t\) \(L_2\): \(x = 3 + 8s\), \(y = -6s\), \(z = 7 + 2s\)
21 Lines and Planes - Line Classification · Level 3
Determine whether the lines \(L_1\) and \(L_2\) are parallel, skew, or intersecting. If they intersect, find the point of intersection. \(L_1\): \(\dfrac{x - 2}{1} = \dfrac{y - 3}{-2} = \dfrac{z - 1}{-3}\) \(L_2\): \(\dfrac{x - 3}{1} = \dfrac{y + 4}{2} = \dfrac{z - 2}{-7}\)
22 Lines and Planes - Line Classification · Level 3
Determine whether the lines \(L_1\) and \(L_2\) are parallel, skew, or intersecting. If they intersect, find the point of intersection. \(L_1\): \(\dfrac{x}{1} = \dfrac{y - 1}{-1} = \dfrac{z - 2}{3}\) \(L_2\): \(\dfrac{x - 2}{2} = \dfrac{y - 3}{-2} = \dfrac{z}{7}\)
23 Lines and Planes - Plane Equations · Level 2
Find an equation of the plane through the origin and perpendicular to the vector \(\langle 1, -2, 5 \rangle\).
24 Lines and Planes - Plane Equations · Level 2
Find an equation of the plane through the point \((5, 3, 5)\) and with normal vector \(2 \mathbf{i} + \mathbf{j} - \mathbf{k}\).
25 Lines and Planes - Plane Equations · Level 2
Find an equation of the plane through the point \(\left(-1, \dfrac{1}{2}, 3\right)\) and with normal vector \(\mathbf{i} + 4 \mathbf{j} + \mathbf{k}\).
26 Lines and Planes - Plane Equations · Level 3
Find an equation of the plane through the point \((2, 0, 1)\) and perpendicular to the line \(x = 3t\), \(y = 2 - t\), \(z = 3 + 4t\).
27 Lines and Planes - Plane Equations · Level 3
Find an equation of the plane through the point \((1, -1, -1)\) and parallel to the plane \(5x - y - z = 6\).
28 Lines and Planes - Plane Equations · Level 3
Find an equation of the plane through the point \((3, -2, 8)\) and parallel to the plane \(z = x + y\).
29 Lines and Planes - Plane Equations · Level 3
Find an equation of the plane through the point \(\left(1, \dfrac{1}{2}, \dfrac{1}{3}\right)\) and parallel to the plane \(x + y + z = 0\).
30 Lines and Planes - Plane Equations · Level 3
Find an equation of the plane that contains the line \(x = 1 + t\), \(y = 2 - t\), \(z = 4 - 3t\) and is parallel to the plane \(5x + 2y + z = 1\).
31 Lines and Planes - Plane Equations · Level 3
Find an equation of the plane through the points \((0, 1, 1)\), \((1, 0, 1)\), and \((1, 1, 0)\).
32 Lines and Planes - Plane Equations · Level 3
Find an equation of the plane through the origin and the points \((3, -2, 1)\) and \((1, 1, 1)\).
33 Lines and Planes - Plane Equations · Level 3
Find an equation of the plane through the points \((2, 1, 2)\), \((3, -8, 6)\), and \((-2, -3, 1)\).
34 Lines and Planes - Plane Equations · Level 3
Find an equation of the plane through the points \((3, 0, -1)\), \((-2, -2, 3)\), and \((7, 1, -4)\).
35 Lines and Planes - Plane Equations · Level 4
Find an equation of the plane that passes through the point \((3, 5, -1)\) and contains the line \(x = 4 - t\), \(y = 2t - 1\), \(z = -3t\).
36 Lines and Planes - Plane Equations · Level 4
Find an equation of the plane that passes through the point \((6, -1, 3)\) and contains the line with symmetric equations \(\dfrac{x}{3} = y + 4 = \dfrac{z}{2}\).
37 Lines and Planes - Plane Equations · Level 4
Find an equation of the plane that passes through the point \((3, 1, 4)\) and contains the line of intersection of the planes \(x + 2y + 3z = 1\) and \(2x - y + z = -3\).
38 Lines and Planes - Plane Equations · Level 4
Find an equation of the plane that passes through the points \((0, -2, 5)\) and \((-1, 3, 1)\) and is perpendicular to the plane \(2z = 5x + 4y\).
39 Lines and Planes - Plane Equations · Level 4
Find an equation of the plane that passes through the point \((1, 5, 1)\) and is perpendicular to the planes \(2x + y - 2z = 2\) and \(x + 3z = 4\).
40 Lines and Planes - Plane Equations · Level 4
Find an equation of the plane that passes through the line of intersection of the planes \(x - z = 1\) and \(y + 2z = 3\) and is perpendicular to the plane \(x + y - 2z = 1\).
41 Lines and Planes - Sketching Planes · Level 2
Use intercepts to help sketch the plane \(2x + 5y + z = 10\).
42 Lines and Planes - Sketching Planes · Level 2
Use intercepts to help sketch the plane \(3x + y + 2z = 6\).
43 Lines and Planes - Sketching Planes · Level 2
Use intercepts to help sketch the plane \(6x - 3y + 4z = 6\).
44 Lines and Planes - Sketching Planes · Level 2
Use intercepts to help sketch the plane \(6x + 5y - 3z = 15\).
45 Lines and Planes - Line-Plane Intersection · Level 3
Find the point at which the line intersects the given plane. \(x = 2 - 2t\), \(y = 3t\), \(z = 1 + t\); \(x + 2y - z = 7\)
46 Lines and Planes - Line-Plane Intersection · Level 3
Find the point at which the line intersects the given plane. \(x = t - 1\), \(y = 1 + 2t\), \(z = 3 - t\); \(3x - y + 2z = 5\)
47 Lines and Planes - Line-Plane Intersection · Level 3
Find the point at which the line intersects the given plane. \(5x = \dfrac{y}{2} = z + 2\); \(10x - 7y + 3z + 24 = 0\)
48 Lines and Planes - Line-Plane Intersection · Level 3
Where does the line through \((-3, 1, 0)\) and \((-1, 5, 6)\) intersect the plane \(2x + y - z = -2\)?
49 Lines and Planes - Plane Intersection · Level 3
Find direction numbers for the line of intersection of the planes \(x + y + z = 1\) and \(x + z = 0\).
50 Lines and Planes - Angle Between Planes · Level 3
Find the cosine of the angle between the planes \(x + y + z = 0\) and \(x + 2y + 3z = 1\).
51 Lines and Planes - Plane Classification · Level 3
Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. \(x + 4y - 3z = 1\), \(\quad -3x + 6y + 7z = 0\)
52 Lines and Planes - Plane Classification · Level 3
Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. \(9x - 3y + 6z = 2\), \(\quad 2y = 6x + 4z\)
53 Lines and Planes - Plane Classification · Level 3
Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. \(x + 2y - z = 2\), \(\quad 2x - 2y + z = 1\)
54 Lines and Planes - Plane Classification · Level 3
Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. \(x - y + 3z = 1\), \(\quad 3x + y - z = 2\)
55 Lines and Planes - Plane Classification · Level 3
Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. \(2x - 3y = z\), \(\quad 4x = 3 + 6y + 2z\)
56 Lines and Planes - Plane Classification · Level 3
Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. \(5x + 2y + 3z = 2\), \(\quad y = 4x - 6z\)
57 Lines and Planes - Plane Intersection · Level 3
(a) Find parametric equations for the line of intersection of the planes \(x + y + z = 1\) and \(x + 2y + 2z = 1\).
(b) Find the angle between the planes.

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58 Lines and Planes - Plane Intersection · Level 3
(a) Find parametric equations for the line of intersection of the planes \(3x - 2y + z = 1\) and \(2x + y - 3z = 3\).
(b) Find the angle between the planes.

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59 Lines and Planes - Plane Intersection · Level 3
Find symmetric equations for the line of intersection of the planes \(5x - 2y - 2z = 1\) and \(4x + y + z = 6\).
60 Lines and Planes - Plane Intersection · Level 3
Find symmetric equations for the line of intersection of the planes \(z = 2x - y - 5\) and \(z = 4x + 3y - 5\).
61 Lines and Planes - Plane Equations · Level 3
Find an equation for the plane consisting of all points that are equidistant from the points \((1, 0, -2)\) and \((3, 4, 0)\).
62 Lines and Planes - Plane Equations · Level 3
Find an equation for the plane consisting of all points that are equidistant from the points \((2, 5, 5)\) and \((-6, 3, 1)\).
63 Lines and Planes - Plane Equations · Level 3
Find an equation of the plane with \(x\)-intercept \(a\), \(y\)-intercept \(b\), and \(z\)-intercept \(c\).
64 Lines and Planes - Line and Plane Equations · Level 4
(a) Find the point at which the given lines intersect: \(\mathbf{r} = \langle 1, 1, 0 \rangle + t \langle 1, -1, 2 \rangle\) \(\mathbf{r} = \langle 2, 0, 2 \rangle + s \langle -1, 1, 0 \rangle\)
(b) Find an equation of the plane that contains these lines.

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65 Lines and Planes - Line and Plane Equations · Level 4
Find parametric equations for the line through the point \((0, 1, 2)\) that is parallel to the plane \(x + y + z = 2\) and perpendicular to the line \(x = 1 + t\), \(y = 1 - t\), \(z = 2t\).
66 Lines and Planes - Line and Plane Equations · Level 4
Find parametric equations for the line through the point \((0, 1, 2)\) that is perpendicular to the line \(x = 1 + t\), \(y = 1 - t\), \(z = 2t\) and intersects this line.
67 Lines and Planes - Parallel Planes · Level 3
Which of the following four planes are parallel? Are any of them identical? \(P_1\): \(3x + 6y - 3z = 6\) \(P_2\): \(4x - 12y + 8z = 5\) \(P_3\): \(9y = 1 + 3x + 6z\) \(P_4\): \(z = x + 2y - 2\)
68 Lines and Planes - Parallel Lines · Level 3
Which of the following four lines are parallel? Are any of them identical? \(L_1\): \(x = 1 + 6t\), \(y = 1 - 3t\), \(z = 12t + 5\) \(L_2\): \(x = 1 + 2t\), \(y = t\), \(z = 1 + 4t\) \(L_3\): \(2x - 2 = 4 - 4y = z + 1\) \(L_4\): \(\mathbf{r} = \langle 3, 1, 5 \rangle + t \langle 4, 2, 8 \rangle\)
69 Lines and Planes - Distance · Level 3
Use the formula in Exercise 12.4.45 to find the distance from the point to the given line. \((4, 1, -2)\); \(\quad x = 1 + t\), \(y = 3 - 2t\), \(z = 4 - 3t\)
70 Lines and Planes - Distance · Level 3
Use the formula in Exercise 12.4.45 to find the distance from the point to the given line. \((0, 1, 3)\); \(\quad x = 2t\), \(y = 6 - 2t\), \(z = 3 + t\)
71 Lines and Planes - Distance · Level 3
Find the distance from the point to the given plane. \((1, -2, 4)\), \(\quad 3x + 2y + 6z = 5\)
72 Lines and Planes - Distance · Level 3
Find the distance from the point to the given plane. \((-6, 3, 5)\), \(\quad x - 2y - 4z = 8\)
73 Lines and Planes - Distance · Level 3
Find the distance between the given parallel planes. \(2x - 3y + z = 4\), \(\quad 4x - 6y + 2z = 3\)
74 Lines and Planes - Distance · Level 3
Find the distance between the given parallel planes. \(6z = 4y - 2x\), \(\quad 9z = 1 - 3x + 6y\)
75 Lines and Planes - Distance Proof · Level 4
Show that the distance between the parallel planes \(a x + b y + c z + d_1 = 0\) and \(a x + b y + c z + d_2 = 0\) is \(D = \dfrac{|d_1 - d_2|}{\sqrt{a^2 + b^2 + c^2}}\)
76 Lines and Planes - Distance · Level 3
Find equations of the planes that are parallel to the plane \(x + 2y - 2z = 1\) and two units away from it.
77 Lines and Planes - Skew Lines Distance · Level 4
Show that the lines with symmetric equations \(x = y = z\) and \(x + 1 = \dfrac{y}{2} = \dfrac{z}{3}\) are skew, and find the distance between these lines.
78 Lines and Planes - Skew Lines Distance · Level 4
Find the distance between the skew lines with parametric equations \(x = 1 + t\), \(y = 1 + 6t\), \(z = 2t\), and \(x = 1 + 2s\), \(y = 5 + 15s\), \(z = -2 + 6s\).
79 Lines and Planes - Skew Lines Distance · Level 4
Let \(L_1\) be the line through the origin and the point \((2, 0, -1)\). Let \(L_2\) be the line through the points \((1, -1, 1)\) and \((4, 1, 3)\). Find the distance between \(L_1\) and \(L_2\).
80 Lines and Planes - Skew Lines Distance · Level 5
Let \(L_1\) be the line through the points \((1, 2, 6)\) and \((2, 4, 8)\). Let \(L_2\) be the line of intersection of the planes \(P_1\) and \(P_2\), where \(P_1\) is the plane \(x - y + 2z + 1 = 0\) and \(P_2\) is the plane through the points \((3, 2, -1)\), \((0, 0, 1)\), and \((1, 2, 1)\). Calculate the distance between \(L_1\) and \(L_2\).
81 Lines and Planes - Application · Level 3
Two tanks are participating in a battle simulation. Tank A is at point \((325, 810, 561)\) and tank B is positioned at point \((765, 675, 599)\).
문제 이미지
(a) Find parametric equations for the line of sight between the tanks.
(b) If we divide the line of sight into 5 equal segments, the elevations of the terrain at the four intermediate points from tank A to tank B are 549, 566, 586, and 589. Can the tanks see each other?

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82 Lines and Planes - Families of Planes · Level 3
Give a geometric description of each family of planes.
(a) \(x + y + z = c\)
(b) \(x + y + c z = 1\)
(c) \(y \cos \theta + z \sin \theta = 1\)

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83 Lines and Planes - Proof · Level 4
If \(a\), \(b\), and \(c\) are not all 0, show that the equation \(a x + b y + c z + d = 0\) represents a plane and \(\langle a, b, c \rangle\) is a normal vector to the plane. Hint: Suppose \(a \neq 0\) and rewrite the equation in the form \(a\left(x + \dfrac{d}{a}\right) + b(y - 0) + c(z - 0) = 0\).

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