Exam 11: Vectors; Lines, Planes, and Quadric Surfaces in Space

The cross product of $\mathbf { v } = - 3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }$ and $\mathbf { w } = 2 \mathbf { i } + \mathbf { j } - \mathbf { k }$ is

(Multiple Choice)

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Question 101

In the three-dimensional Euclidean space, the graph of $x ^ { 2 } + \frac { y ^ { 2 } } { 4 } - \frac { z ^ { 2 } } { 9 } = 1$ is

(Multiple Choice)

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Question 102

The equation of the plane through (5,1,3) parallel to the yz-plane is

(Multiple Choice)

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Question 103

The cross product of $\mathbf { v } = - 3 \mathbf { i } + \mathbf { j } - \mathbf { k }$ and $\mathbf { w } = 2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k }$ is

(Multiple Choice)

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Question 104

The cross product of $\mathbf { v } = 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k }$ and $\mathbf { w } = 3 \mathbf { i } + 6 \mathbf { j } - 2 \mathbf { k }$ is

(Multiple Choice)

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Question 105

In the three-dimensional Euclidean space, the graph of $z ^ { 2 } = x ^ { 2 } + \frac { y ^ { 2 } } { 4 }$ is

(Multiple Choice)

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Question 106

The vector projection of $\mathbf { v } = 2 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }$ onto $\mathbf { w } = \mathbf { i } - \mathbf { j } - \mathbf { k }$ is

(Multiple Choice)

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Question 107

The area of the parallelogram PQRS with vertices $P = ( 0,0,3 ) , Q = ( 2 , - 5,3 )$ and $R = ( 1,1 , - 2 )$ is

(Multiple Choice)

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Question 108

Which of the following is not true?

(Multiple Choice)

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Question 109

In the three-dimensional Euclidean space, the graph of $\frac { x ^ { 2 } } { 4 } + y ^ { 2 } - \frac { z ^ { 2 } } { 16 } = - 1$ is

(Multiple Choice)

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Question 110

Which of the following is not true?

(Multiple Choice)

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Question 111

The parametric equations of the line through the points (3,2,1) and (2,4,1) is

(Multiple Choice)

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Question 112

Let , and $\mathbf { u } = \langle 1,1 , - 6 , \rangle$ Then $\mathbf { v } = \langle 3,0,8 \rangle$ is

(Multiple Choice)

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Question 113

In the three-dimensional Euclidean space, the graph of $x ^ { 2 } + y ^ { 2 } = 36$ is

(Multiple Choice)

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Question 114

In the three-dimensional Euclidean space, the graph of $z ^ { 2 } = \frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 16 }$ is

(Multiple Choice)

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Question 115

The symmetric equations of the line through (3,-2,5) in the direction of $\frac { x - 3 } { 3 } = \frac { y - 2 } { - 1 } = \frac { z - 1 } { - 1 }$ is

(Multiple Choice)

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Question 116

Which of the following is not true?

(Multiple Choice)

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Question 117

In the three-dimensional Euclidean space, the graph of $z = \frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 9 }$ is

(Multiple Choice)

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Question 118

The area of the parallelogram PQRS with vertices $P = ( - 1,1,1 ) , Q = ( - 1,2,2 )$ and $R = ( - 3,5 , - 4 )$ is

(Multiple Choice)

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Question 119

The value of a that makes $\mathbf { v } = 3 \mathbf { i } - 4 \mathbf { j } + a \mathbf { k }$ and $\mathbf { w } = 2 \mathbf { i } - \mathbf { j } + 5 \mathbf { k }$ orthogonal is

(Multiple Choice)

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Question 120

The cross product of $\mathbf { v } = - 3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }$ and $\mathbf { w } = 2 \mathbf { i } + \mathbf { j } - \mathbf { k }$ is

In the three-dimensional Euclidean space, the graph of $x ^ { 2 } + \frac { y ^ { 2 } } { 4 } - \frac { z ^ { 2 } } { 9 } = 1$ is

The equation of the plane through (5,1,3) parallel to the yz-plane is

The cross product of $\mathbf { v } = - 3 \mathbf { i } + \mathbf { j } - \mathbf { k }$ and $\mathbf { w } = 2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k }$ is

The cross product of $\mathbf { v } = 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k }$ and $\mathbf { w } = 3 \mathbf { i } + 6 \mathbf { j } - 2 \mathbf { k }$ is

In the three-dimensional Euclidean space, the graph of $z ^ { 2 } = x ^ { 2 } + \frac { y ^ { 2 } } { 4 }$ is

The vector projection of $\mathbf { v } = 2 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }$ onto $\mathbf { w } = \mathbf { i } - \mathbf { j } - \mathbf { k }$ is

The area of the parallelogram PQRS with vertices $P = ( 0,0,3 ) , Q = ( 2 , - 5,3 )$ and $R = ( 1,1 , - 2 )$ is

Which of the following is not true?

In the three-dimensional Euclidean space, the graph of $\frac { x ^ { 2 } } { 4 } + y ^ { 2 } - \frac { z ^ { 2 } } { 16 } = - 1$ is

Which of the following is not true?

The parametric equations of the line through the points (3,2,1) and (2,4,1) is

Let , and $\mathbf { u } = \langle 1,1 , - 6 , \rangle$ Then $\mathbf { v } = \langle 3,0,8 \rangle$ is

In the three-dimensional Euclidean space, the graph of $x ^ { 2 } + y ^ { 2 } = 36$ is

In the three-dimensional Euclidean space, the graph of $z ^ { 2 } = \frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 16 }$ is

The symmetric equations of the line through (3,-2,5) in the direction of $\frac { x - 3 } { 3 } = \frac { y - 2 } { - 1 } = \frac { z - 1 } { - 1 }$ is

Which of the following is not true?

In the three-dimensional Euclidean space, the graph of $z = \frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 9 }$ is

The area of the parallelogram PQRS with vertices $P = ( - 1,1,1 ) , Q = ( - 1,2,2 )$ and $R = ( - 3,5 , - 4 )$ is

The value of a that makes $\mathbf { v } = 3 \mathbf { i } - 4 \mathbf { j } + a \mathbf { k }$ and $\mathbf { w } = 2 \mathbf { i } - \mathbf { j } + 5 \mathbf { k }$ orthogonal is

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Exam 11: Vectors; Lines, Planes, and Quadric Surfaces in Space

The cross product of v=−3i+2j−k\mathbf { v } = - 3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }v=−3i+2j−k and w=2i+j−k\mathbf { w } = 2 \mathbf { i } + \mathbf { j } - \mathbf { k }w=2i+j−k is

In the three-dimensional Euclidean space, the graph of x2+y24−z29=1x ^ { 2 } + \frac { y ^ { 2 } } { 4 } - \frac { z ^ { 2 } } { 9 } = 1x2+4y2​−9z2​=1 is

The equation of the plane through (5,1,3) parallel to the yz-plane is

The cross product of v=−3i+j−k\mathbf { v } = - 3 \mathbf { i } + \mathbf { j } - \mathbf { k }v=−3i+j−k and w=2i+j−3k\mathbf { w } = 2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k }w=2i+j−3k is

The cross product of v=2i+j−2k\mathbf { v } = 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k }v=2i+j−2k and w=3i+6j−2k\mathbf { w } = 3 \mathbf { i } + 6 \mathbf { j } - 2 \mathbf { k }w=3i+6j−2k is

In the three-dimensional Euclidean space, the graph of z2=x2+y24z ^ { 2 } = x ^ { 2 } + \frac { y ^ { 2 } } { 4 }z2=x2+4y2​ is

The vector projection of v=2i+3j+k\mathbf { v } = 2 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }v=2i+3j+k onto w=i−j−k\mathbf { w } = \mathbf { i } - \mathbf { j } - \mathbf { k }w=i−j−k is

The area of the parallelogram PQRS with vertices P=(0,0,3),Q=(2,−5,3)P = ( 0,0,3 ) , Q = ( 2 , - 5,3 )P=(0,0,3),Q=(2,−5,3) and R=(1,1,−2)R = ( 1,1 , - 2 )R=(1,1,−2) is

Which of the following is not true?

In the three-dimensional Euclidean space, the graph of x24+y2−z216=−1\frac { x ^ { 2 } } { 4 } + y ^ { 2 } - \frac { z ^ { 2 } } { 16 } = - 14x2​+y2−16z2​=−1 is

Which of the following is not true? ​

The parametric equations of the line through the points (3,2,1) and (2,4,1) is

Let , and u=⟨1,1,−6,⟩\mathbf { u } = \langle 1,1 , - 6 , \rangleu=⟨1,1,−6,⟩ Then v=⟨3,0,8⟩\mathbf { v } = \langle 3,0,8 \ranglev=⟨3,0,8⟩ is

In the three-dimensional Euclidean space, the graph of x2+y2=36x ^ { 2 } + y ^ { 2 } = 36x2+y2=36 is

In the three-dimensional Euclidean space, the graph of z2=x24+y216z ^ { 2 } = \frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 16 }z2=4x2​+16y2​ is

The symmetric equations of the line through (3,-2,5) in the direction of x−33=y−2−1=z−1−1\frac { x - 3 } { 3 } = \frac { y - 2 } { - 1 } = \frac { z - 1 } { - 1 }3x−3​=−1y−2​=−1z−1​ is

Which of the following is not true? ​

In the three-dimensional Euclidean space, the graph of z=x24−y29z = \frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 9 }z=4x2​−9y2​ is

The area of the parallelogram PQRS with vertices P=(−1,1,1),Q=(−1,2,2)P = ( - 1,1,1 ) , Q = ( - 1,2,2 )P=(−1,1,1),Q=(−1,2,2) and R=(−3,5,−4)R = ( - 3,5 , - 4 )R=(−3,5,−4) is

The value of a that makes v=3i−4j+ak\mathbf { v } = 3 \mathbf { i } - 4 \mathbf { j } + a \mathbf { k }v=3i−4j+ak and w=2i−j+5k\mathbf { w } = 2 \mathbf { i } - \mathbf { j } + 5 \mathbf { k }w=2i−j+5k orthogonal is

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The Derivative

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Applications of the Derivative

The Integral

Applications of the Integral

Techniques of Integration

Infinite Series

Parametric Equations; Polar Equations

Vector Functions

Functions of Several Variables

Directional Derivatives, Gradients, and Extrema

Multiple Integrals

Vector Calculus

Differential Equations

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The cross product of $\mathbf { v } = - 3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }$ and $\mathbf { w } = 2 \mathbf { i } + \mathbf { j } - \mathbf { k }$ is

In the three-dimensional Euclidean space, the graph of $x ^ { 2 } + \frac { y ^ { 2 } } { 4 } - \frac { z ^ { 2 } } { 9 } = 1$ is

The cross product of $\mathbf { v } = - 3 \mathbf { i } + \mathbf { j } - \mathbf { k }$ and $\mathbf { w } = 2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k }$ is

The cross product of $\mathbf { v } = 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k }$ and $\mathbf { w } = 3 \mathbf { i } + 6 \mathbf { j } - 2 \mathbf { k }$ is

In the three-dimensional Euclidean space, the graph of $z ^ { 2 } = x ^ { 2 } + \frac { y ^ { 2 } } { 4 }$ is

The vector projection of $\mathbf { v } = 2 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }$ onto $\mathbf { w } = \mathbf { i } - \mathbf { j } - \mathbf { k }$ is

The area of the parallelogram PQRS with vertices $P = ( 0,0,3 ) , Q = ( 2 , - 5,3 )$ and $R = ( 1,1 , - 2 )$ is

In the three-dimensional Euclidean space, the graph of $\frac { x ^ { 2 } } { 4 } + y ^ { 2 } - \frac { z ^ { 2 } } { 16 } = - 1$ is

Which of the following is not true?

Let , and $\mathbf { u } = \langle 1,1 , - 6 , \rangle$ Then $\mathbf { v } = \langle 3,0,8 \rangle$ is

In the three-dimensional Euclidean space, the graph of $x ^ { 2 } + y ^ { 2 } = 36$ is

In the three-dimensional Euclidean space, the graph of $z ^ { 2 } = \frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 16 }$ is

The symmetric equations of the line through (3,-2,5) in the direction of $\frac { x - 3 } { 3 } = \frac { y - 2 } { - 1 } = \frac { z - 1 } { - 1 }$ is

Which of the following is not true?

In the three-dimensional Euclidean space, the graph of $z = \frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 9 }$ is

The area of the parallelogram PQRS with vertices $P = ( - 1,1,1 ) , Q = ( - 1,2,2 )$ and $R = ( - 3,5 , - 4 )$ is

The value of a that makes $\mathbf { v } = 3 \mathbf { i } - 4 \mathbf { j } + a \mathbf { k }$ and $\mathbf { w } = 2 \mathbf { i } - \mathbf { j } + 5 \mathbf { k }$ orthogonal is