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

The angle, in degrees and correct to two decimal places, between $\mathbf { v } = \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k }$ and $\mathbf { w } = 3 \mathbf { i } + \mathbf { j } - 3 \mathbf { k }$ is

(Multiple Choice)

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Question 121

In the three-dimensional Euclidean space, the graph of $y = 3 x ^ { 2 }$ is

(Multiple Choice)

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Question 122

Which of the following is not true?

(Multiple Choice)

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Question 123

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 124

Which of the following is true?

(Multiple Choice)

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Question 125

The angle, in degrees and correct to two decimal places, between $\mathbf { v } = - 3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }$ and $\mathbf { w } = 2 \mathbf { i } + \mathbf { j } - \mathbf { k }$ is

(Multiple Choice)

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Question 126

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

(Multiple Choice)

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Question 127

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

(Multiple Choice)

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Question 128

The distance between $P _ { 1 } = ( - 1,2 , - 6 )$ and $P _ { 2 } = ( 1,2 , - 5 )$ is

(Multiple Choice)

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Question 129

The area of the parallelogram PQRS with vertices $P = ( 1,1 , - 6 ) , Q = ( 5 , - 3,0 ) \text {, }$ and $R = ( - 2,4,1 )$ is

(Multiple Choice)

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Question 130

The distance between $P _ { 1 } = ( 10 , - 3,4 )$ and $P _ { 2 } = ( 8,2,6 )$ is

(Multiple Choice)

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Question 131

The unit vector resulted from normalizing the vector $\mathbf { w } = 2 \mathbf { i } - 3 \mathbf { j } - 4 \mathbf { k }$ is

(Multiple Choice)

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Question 132

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

(Multiple Choice)

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Question 133

The volume of the parallelepiped whose adjacent sides are the vectors and $\mathbf { u } = 8 \mathbf { i } - 6 \mathbf { j } + 5 \mathbf { k }$ is

(Multiple Choice)

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Question 134

The distance between $P _ { 1 } = ( 1,2,3 )$ and $P _ { 2 } = ( 4,5,6 )$ is

(Multiple Choice)

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Question 135

Let $P _ { 1 } = ( - 2,4 , - 7 )$ and $P _ { 2 } = ( 3,1 , - 6 )$ If v is $\overrightarrow { P _ { 1 } P _ { 2 } }$ in standard position, then v is

(Multiple Choice)

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Question 136

The unit vector resulted from normalizing the vector $\mathbf { w } = \mathbf { i } + \mathbf { j } + \mathbf { k }$ is

(Multiple Choice)

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Question 137

The center and radius of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 2 x - 2 z = - 1$ are

(Multiple Choice)

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Question 138

The angle, in degrees and correct to two decimal places, between $\mathbf { v } = \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k }$ and $\mathbf { w } = 3 \mathbf { i } + \mathbf { j } - 3 \mathbf { k }$ is

In the three-dimensional Euclidean space, the graph of $y = 3 x ^ { 2 }$ 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

Which of the following is true?

The angle, in degrees and correct to two decimal places, between $\mathbf { v } = - 3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }$ and $\mathbf { w } = 2 \mathbf { i } + \mathbf { j } - \mathbf { k }$ is

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

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

The distance between $P _ { 1 } = ( - 1,2 , - 6 )$ and $P _ { 2 } = ( 1,2 , - 5 )$ is

The area of the parallelogram PQRS with vertices $P = ( 1,1 , - 6 ) , Q = ( 5 , - 3,0 ) \text {, }$ and $R = ( - 2,4,1 )$ is

The distance between $P _ { 1 } = ( 10 , - 3,4 )$ and $P _ { 2 } = ( 8,2,6 )$ is

The unit vector resulted from normalizing the vector $\mathbf { w } = 2 \mathbf { i } - 3 \mathbf { j } - 4 \mathbf { k }$ is

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

The volume of the parallelepiped whose adjacent sides are the vectors and $\mathbf { u } = 8 \mathbf { i } - 6 \mathbf { j } + 5 \mathbf { k }$ is

The distance between $P _ { 1 } = ( 1,2,3 )$ and $P _ { 2 } = ( 4,5,6 )$ is

Let $P _ { 1 } = ( - 2,4 , - 7 )$ and $P _ { 2 } = ( 3,1 , - 6 )$ If v is $\overrightarrow { P _ { 1 } P _ { 2 } }$ in standard position, then v is

The unit vector resulted from normalizing the vector $\mathbf { w } = \mathbf { i } + \mathbf { j } + \mathbf { k }$ is

The center and radius of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 2 x - 2 z = - 1$ are

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

The angle, in degrees and correct to two decimal places, between v=i−3j+4k\mathbf { v } = \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k }v=i−3j+4k and w=3i+j−3k\mathbf { w } = 3 \mathbf { i } + \mathbf { j } - 3 \mathbf { k }w=3i+j−3k is

In the three-dimensional Euclidean space, the graph of y=3x2y = 3 x ^ { 2 }y=3x2 is

Which of the following is not true? ​

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

Which of the following is true? ​

The angle, in degrees and correct to two decimal places, between 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

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+k\mathbf { w } = - 2 \mathbf { i } + \mathbf { j } + \mathbf { k }w=−2i+j+k is

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

The distance between P1=(−1,2,−6)P _ { 1 } = ( - 1,2 , - 6 )P1​=(−1,2,−6) and P2=(1,2,−5)P _ { 2 } = ( 1,2 , - 5 )P2​=(1,2,−5) is

The area of the parallelogram PQRS with vertices P=(1,1,−6),Q=(5,−3,0), P = ( 1,1 , - 6 ) , Q = ( 5 , - 3,0 ) \text {, }P=(1,1,−6),Q=(5,−3,0), and R=(−2,4,1)R = ( - 2,4,1 )R=(−2,4,1) is

The distance between P1=(10,−3,4)P _ { 1 } = ( 10 , - 3,4 )P1​=(10,−3,4) and P2=(8,2,6)P _ { 2 } = ( 8,2,6 )P2​=(8,2,6) is

The unit vector resulted from normalizing the vector w=2i−3j−4k\mathbf { w } = 2 \mathbf { i } - 3 \mathbf { j } - 4 \mathbf { k }w=2i−3j−4k is

In the three-dimensional Euclidean space, the graph of z=x2+4y2z = x ^ { 2 } + 4 y ^ { 2 }z=x2+4y2 is

The volume of the parallelepiped whose adjacent sides are the vectors and u=8i−6j+5k\mathbf { u } = 8 \mathbf { i } - 6 \mathbf { j } + 5 \mathbf { k }u=8i−6j+5k is

The distance between P1=(1,2,3)P _ { 1 } = ( 1,2,3 )P1​=(1,2,3) and P2=(4,5,6)P _ { 2 } = ( 4,5,6 )P2​=(4,5,6) is

Let P1=(−2,4,−7)P _ { 1 } = ( - 2,4 , - 7 )P1​=(−2,4,−7) and P2=(3,1,−6)P _ { 2 } = ( 3,1 , - 6 )P2​=(3,1,−6) If v is P1P2→\overrightarrow { P _ { 1 } P _ { 2 } }P1​P2​​ in standard position, then v is

The unit vector resulted from normalizing the vector w=i+j+k\mathbf { w } = \mathbf { i } + \mathbf { j } + \mathbf { k }w=i+j+k is

The center and radius of the sphere x2+y2+z2+2x−2z=−1x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 2 x - 2 z = - 1x2+y2+z2+2x−2z=−1 are

Preparing for Calculus

Limits and Continuity

The Derivative

More About Derivatives

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

Filters

The angle, in degrees and correct to two decimal places, between $\mathbf { v } = \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k }$ and $\mathbf { w } = 3 \mathbf { i } + \mathbf { j } - 3 \mathbf { k }$ is

In the three-dimensional Euclidean space, the graph of $y = 3 x ^ { 2 }$ 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

Which of the following is true?

The angle, in degrees and correct to two decimal places, between $\mathbf { v } = - 3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }$ and $\mathbf { w } = 2 \mathbf { i } + \mathbf { j } - \mathbf { k }$ is

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

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

The distance between $P _ { 1 } = ( - 1,2 , - 6 )$ and $P _ { 2 } = ( 1,2 , - 5 )$ is

The area of the parallelogram PQRS with vertices $P = ( 1,1 , - 6 ) , Q = ( 5 , - 3,0 ) \text {, }$ and $R = ( - 2,4,1 )$ is

The distance between $P _ { 1 } = ( 10 , - 3,4 )$ and $P _ { 2 } = ( 8,2,6 )$ is

The unit vector resulted from normalizing the vector $\mathbf { w } = 2 \mathbf { i } - 3 \mathbf { j } - 4 \mathbf { k }$ is

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

The volume of the parallelepiped whose adjacent sides are the vectors and $\mathbf { u } = 8 \mathbf { i } - 6 \mathbf { j } + 5 \mathbf { k }$ is

The distance between $P _ { 1 } = ( 1,2,3 )$ and $P _ { 2 } = ( 4,5,6 )$ is

Let $P _ { 1 } = ( - 2,4 , - 7 )$ and $P _ { 2 } = ( 3,1 , - 6 )$ If v is $\overrightarrow { P _ { 1 } P _ { 2 } }$ in standard position, then v is

The unit vector resulted from normalizing the vector $\mathbf { w } = \mathbf { i } + \mathbf { j } + \mathbf { k }$ is

The center and radius of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 2 x - 2 z = - 1$ are