Question 1

Let L be the line given by x = 2 - t, y = 1 + t, and z = 1 + 2t. L intersects the plane 2x + y - z = 1 at the point P = (1, 2, 3). Find the angle L makes with the plane, to the nearest degree.

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Question 2

Let l and $ l^{\prime} $ be two lines in space given by the equations
x = 3 + t
x = -1 + t
$ l $ :
y = 1 - t $ l^{\prime} $ :
y = 2t
z = 2t
z = 1 + kt
Find all values of k (if any) for which l and $l ^ { \prime }$ are perpendicular.

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The answer of Let l and $ l^{\prime} $ be...

Question 3

Find the intersections of the line passing through the points (-1, 3, 4) and (3, 5, 2) with the yz-plane, the xz-plane, and the xy-plane.

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Question 4

Find the equation of the plane containing the lines x = 4 - 4t, y = 3 - t, z = 1 + 5t and x = 4 - t, y = 3 + 2t, z = 1.

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The answer of Find the equation of the plane containing...

Question 5

Find the distance between the two skew lines $L _ { 1 } : \frac { x - 1 } { 2 } = \frac { y - 3 } { 5 } = \frac { z - 1 } { 3 }$ and $L _ { 2 } : \frac { x - 2 } { 4 } = \frac { y + 1 } { 2 } = \frac { z + 2 } { 3 }$

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Question 6

Find equations of all planes containing the x-axis.

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Question 7

Find the angle between the lines $\frac { x - 2 } { 1 } = \frac { 1 - y } { 3 } = \frac { z - 3 } { 1 }$ and $\frac { x } { 2 } = \frac { y + 3 } { - 1 } = \frac { z - 1 } { 2 }$ .

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Question 8

Are the planes 3x - y + 5z = 13 and x + 7y - 2z = 4 perpendicular to each other?

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Question 9

Find equations of all planes containing z-axis.

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Question 10

Determine whether the lines L₁: x = 1 + 7t, y = 3 + t, z = 5 - 3t and L₂: x = 4 - t, y = 4, z = 7 + 2t are parallel, intersecting or skew. If they intersect, find the point of intersection.

Accepted Answer

The answer of Determine whether the lines L₁: x =...

Question 11

Let l and $l ^ { \prime }$ be two lines in space given by the equations
x = 3 + t
x = -1 + t
l:
y = 1 - t $l ^ { \prime }$ :
y = 2t
z = 2t
z = 1 + kt
Find all values of k (if any) for which l and $l ^ { \prime }$ are parallel.

Accepted Answer

The answer of Let l and \(l ^ { \prime...

Question 12

Find equations of all planes containing the y-axis.

Accepted Answer

The answer of Find equations of all planes containing the...

Question 13

Find the intersection point of the line $\mathbf { r } = \langle 1,0,2 \rangle + \langle 2 , - 2,1 \rangle t$ and the plane 3x + 4y + 6z = 7.

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The answer of Find the intersection point of the line...

Question 14

Determine whether the lines L₁: x = 1 + t, y = 2 + 3t, z = 3 + t and L₂: x = 1 + t, y = 3 + 4t, z = 4 + 2t are parallel, intersecting or skew. If they intersect, find the point of intersection.

Accepted Answer

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Question 15

Find parametric equations and symmetric equations of the line passing through the points A and B shown below.

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Question 16

Find parametric equations and symmetric equations of the line passing through the points A and B shown below.

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Question 17

Find an equation of the plane through the point P = (2, 1, -4) and perpendicular to the line x = 2 + 3t, y = 1 - 4t, z = 3 + 3t.

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Question 18

Let L be the line given by x = 2 - t, y = 1 + t, and z = 1 + 2t. L intersects the plane 2x + y - z = 1 at the point P = (1, 2, 3). Find parametric equations for the line through P which lies in the plane and is perpendicular to L.

Accepted Answer

The answer of Let L be the line given by...

Question 19

Given the points A = (2, 3, 1), B = (4, -1, 5), and O = (0, 0, 0), find the distance from O to the line through A and B.

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Question 20

Find the equation of the plane containing the points (5, 3, 1), (1, 8, 4), and (-1, 3, -2).

Accepted Answer

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Quiz 9: Vectors and the Geometry of Space

Let L be the line given by x = 2 - t, y = 1 + t, and z = 1 + 2t. L intersects the plane 2x + y - z = 1 at the point P = (1, 2, 3). Find the angle L makes with the plane, to the nearest degree.

Let l and $l^{\prime}$ be two lines in space given by the equations x = 3 + t x = -1 + t $l$ : y = 1 - t $l^{\prime}$ : y = 2t z = 2t z = 1 + kt Find all values of k (if any) for which l and $l ^ { \prime }$ are perpendicular.

Find the intersections of the line passing through the points (-1, 3, 4) and (3, 5, 2) with the yz-plane, the xz-plane, and the xy-plane.

Find the equation of the plane containing the lines x = 4 - 4t, y = 3 - t, z = 1 + 5t and x = 4 - t, y = 3 + 2t, z = 1.

Find the distance between the two skew lines $L _ { 1 } : \frac { x - 1 } { 2 } = \frac { y - 3 } { 5 } = \frac { z - 1 } { 3 }$ and $L _ { 2 } : \frac { x - 2 } { 4 } = \frac { y + 1 } { 2 } = \frac { z + 2 } { 3 }$

Find equations of all planes containing the x-axis.

Find the angle between the lines $\frac { x - 2 } { 1 } = \frac { 1 - y } { 3 } = \frac { z - 3 } { 1 }$ and $\frac { x } { 2 } = \frac { y + 3 } { - 1 } = \frac { z - 1 } { 2 }$ .

Are the planes 3x - y + 5z = 13 and x + 7y - 2z = 4 perpendicular to each other?

Find equations of all planes containing z-axis.

Determine whether the lines L₁: x = 1 + 7t, y = 3 + t, z = 5 - 3t and L₂: x = 4 - t, y = 4, z = 7 + 2t are parallel, intersecting or skew. If they intersect, find the point of intersection.

Let l and $l ^ { \prime }$ be two lines in space given by the equations x = 3 + t x = -1 + t l: y = 1 - t $l ^ { \prime }$ : y = 2t z = 2t z = 1 + kt Find all values of k (if any) for which l and $l ^ { \prime }$ are parallel.

Find equations of all planes containing the y-axis.

Find the intersection point of the line $\mathbf { r } = \langle 1,0,2 \rangle + \langle 2 , - 2,1 \rangle t$ and the plane 3x + 4y + 6z = 7.

Determine whether the lines L₁: x = 1 + t, y = 2 + 3t, z = 3 + t and L₂: x = 1 + t, y = 3 + 4t, z = 4 + 2t are parallel, intersecting or skew. If they intersect, find the point of intersection.

Find parametric equations and symmetric equations of the line passing through the points A and B shown below.

Find parametric equations and symmetric equations of the line passing through the points A and B shown below.

Find an equation of the plane through the point P = (2, 1, -4) and perpendicular to the line x = 2 + 3t, y = 1 - 4t, z = 3 + 3t.

Let L be the line given by x = 2 - t, y = 1 + t, and z = 1 + 2t. L intersects the plane 2x + y - z = 1 at the point P = (1, 2, 3). Find parametric equations for the line through P which lies in the plane and is perpendicular to L.

Given the points A = (2, 3, 1), B = (4, -1, 5), and O = (0, 0, 0), find the distance from O to the line through A and B.

Find the equation of the plane containing the points (5, 3, 1), (1, 8, 4), and (-1, 3, -2).

Quiz 9: Vectors and the Geometry of Space

Let L be the line given by x = 2 - t, y = 1 + t, and z = 1 + 2t. L intersects the plane 2x + y - z = 1 at the point P = (1, 2, 3). Find the angle L makes with the plane, to the nearest degree.

Let l and l′ l^{\prime} l′ be two lines in space given by the equations x = 3 + t x = -1 + t l l l : y = 1 - t l′ l^{\prime} l′ : y = 2t z = 2t z = 1 + kt Find all values of k (if any) for which l and l′l ^ { \prime }l′ are perpendicular.

Find the intersections of the line passing through the points (-1, 3, 4) and (3, 5, 2) with the yz-plane, the xz-plane, and the xy-plane.

Find the equation of the plane containing the lines x = 4 - 4t, y = 3 - t, z = 1 + 5t and x = 4 - t, y = 3 + 2t, z = 1.

Find equations of all planes containing the x-axis.

Find the angle between the lines x−21=1−y3=z−31\frac { x - 2 } { 1 } = \frac { 1 - y } { 3 } = \frac { z - 3 } { 1 }1x−2​=31−y​=1z−3​ and x2=y+3−1=z−12\frac { x } { 2 } = \frac { y + 3 } { - 1 } = \frac { z - 1 } { 2 }2x​=−1y+3​=2z−1​ .

Are the planes 3x - y + 5z = 13 and x + 7y - 2z = 4 perpendicular to each other?

Find equations of all planes containing z-axis.

Determine whether the lines L1: x = 1 + 7t, y = 3 + t, z = 5 - 3t and L2: x = 4 - t, y = 4, z = 7 + 2t are parallel, intersecting or skew. If they intersect, find the point of intersection.

Let l and l′l ^ { \prime }l′ be two lines in space given by the equations x = 3 + t x = -1 + t l: y = 1 - t l′l ^ { \prime }l′ : y = 2t z = 2t z = 1 + kt Find all values of k (if any) for which l and l′l ^ { \prime }l′ are parallel.

Find equations of all planes containing the y-axis.

Find the intersection point of the line r=⟨1,0,2⟩+⟨2,−2,1⟩t\mathbf { r } = \langle 1,0,2 \rangle + \langle 2 , - 2,1 \rangle tr=⟨1,0,2⟩+⟨2,−2,1⟩t and the plane 3x + 4y + 6z = 7.

Determine whether the lines L1: x = 1 + t, y = 2 + 3t, z = 3 + t and L2: x = 1 + t, y = 3 + 4t, z = 4 + 2t are parallel, intersecting or skew. If they intersect, find the point of intersection.

Find parametric equations and symmetric equations of the line passing through the points A and B shown below.

Find parametric equations and symmetric equations of the line passing through the points A and B shown below.

Find an equation of the plane through the point P = (2, 1, -4) and perpendicular to the line x = 2 + 3t, y = 1 - 4t, z = 3 + 3t.

Let L be the line given by x = 2 - t, y = 1 + t, and z = 1 + 2t. L intersects the plane 2x + y - z = 1 at the point P = (1, 2, 3). Find parametric equations for the line through P which lies in the plane and is perpendicular to L.

Given the points A = (2, 3, 1), B = (4, -1, 5), and O = (0, 0, 0), find the distance from O to the line through A and B.

Find the equation of the plane containing the points (5, 3, 1), (1, 8, 4), and (-1, 3, -2).

Let l and $l^{\prime}$ be two lines in space given by the equations x = 3 + t x = -1 + t $l$ : y = 1 - t $l^{\prime}$ : y = 2t z = 2t z = 1 + kt Find all values of k (if any) for which l and $l ^ { \prime }$ are perpendicular.

Find the angle between the lines $\frac { x - 2 } { 1 } = \frac { 1 - y } { 3 } = \frac { z - 3 } { 1 }$ and $\frac { x } { 2 } = \frac { y + 3 } { - 1 } = \frac { z - 1 } { 2 }$ .

Determine whether the lines L₁: x = 1 + 7t, y = 3 + t, z = 5 - 3t and L₂: x = 4 - t, y = 4, z = 7 + 2t are parallel, intersecting or skew. If they intersect, find the point of intersection.

Let l and $l ^ { \prime }$ be two lines in space given by the equations x = 3 + t x = -1 + t l: y = 1 - t $l ^ { \prime }$ : y = 2t z = 2t z = 1 + kt Find all values of k (if any) for which l and $l ^ { \prime }$ are parallel.

Find the intersection point of the line $\mathbf { r } = \langle 1,0,2 \rangle + \langle 2 , - 2,1 \rangle t$ and the plane 3x + 4y + 6z = 7.

Determine whether the lines L₁: x = 1 + t, y = 2 + 3t, z = 3 + t and L₂: x = 1 + t, y = 3 + 4t, z = 4 + 2t are parallel, intersecting or skew. If they intersect, find the point of intersection.