Question 1

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

Accepted Answer

A) An elliptic cylinder 
B) A hyperboloid of one sheet 
C) A circular cylinder 
D) A hyperboloid of two sheets 
E) A parabolic cylinder 
A) An elliptic cylinder 
B) A hyperboloid of one sheet 
C) A circular cylinder 
D) A hyperboloid of two sheets 
E) A parabolic cylinder

Question 2

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

Accepted Answer

A)  $\langle - 21,1 , - 42 \rangle$ 
B)  $\langle - 21 , - 1 , - 42 \rangle$ 
C)  $\langle - 21 , - 1,42 \rangle$ 
D)  $\langle 21 , - 1,42 \rangle$ 
E)  $\langle 21 , - 1 , - 42 \rangle$ 
A)  $\langle - 21,1 , - 42 \rangle$ 
B)  $\langle - 21 , - 1 , - 42 \rangle$ 
C)  $\langle - 21 , - 1,42 \rangle$ 
D)  $\langle 21 , - 1,42 \rangle$ 
E)  $\langle 21 , - 1 , - 42 \rangle$

Question 3

The equation of the sphere centered at (7,-1,3) and tangent to the yz-plane is

Accepted Answer

A)  $( x - 7 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + ( z - 3 ) ^ { 2 } = 49$ 
B)  $( x + 7 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + ( z - 3 ) ^ { 2 } = 49$ 
C)  $( x - 7 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z - 3 ) ^ { 2 } = 36$ 
D)  $( x - 7 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z - 3 ) ^ { 2 } = 49$ 
E)  $( x - 1 ) ^ { 2 } + ( y - 5 ) ^ { 2 } + ( z + 3 ) ^ { 2 } = 36$ 
A)  $( x - 7 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + ( z - 3 ) ^ { 2 } = 49$ 
B)  $( x + 7 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + ( z - 3 ) ^ { 2 } = 49$ 
C)  $( x - 7 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z - 3 ) ^ { 2 } = 36$ 
D)  $( x - 7 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z - 3 ) ^ { 2 } = 49$ 
E)  $( x - 1 ) ^ { 2 } + ( y - 5 ) ^ { 2 } + ( z + 3 ) ^ { 2 } = 36$

Question 4

Which of the following is not true?
​

Accepted Answer

A) The sum of two scalars is a scalar. 
B) Vector addition is commutative. 
C) Scalar addition is commutative. 
D) Scalar multiplication is commutative. 
E) The difference of a vector and a scalar is a vector. 
A) The sum of two scalars is a scalar. 
B) Vector addition is commutative. 
C) Scalar addition is commutative. 
D) Scalar multiplication is commutative. 
E) The difference of a vector and a scalar is a vector.

Question 5

The equation of the plane through (1,-1,2) normal to $2 \mathbf { i } - \mathbf { j } + \mathbf { k }$ is

Accepted Answer

A)  $2 x - y + z = 5$ 
B)  $2 x - y + z = 4$ 
C)  $2 x - y + z = 2$ 
D)  $2 x - y + z = - 3$ 
E)  $2 x - y + z = 3$ 
A)  $2 x - y + z = 5$ 
B)  $2 x - y + z = 4$ 
C)  $2 x - y + z = 2$ 
D)  $2 x - y + z = - 3$ 
E)  $2 x - y + z = 3$

Question 6

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

Accepted Answer

A) An elliptic cylinder 
B) A hyperboloid of one sheet 
C) A circular cylinder 
D) A hyperboloid of two sheets 
E) A parabolic cylinder 
A) An elliptic cylinder 
B) A hyperboloid of one sheet 
C) A circular cylinder 
D) A hyperboloid of two sheets 
E) A parabolic cylinder

Question 7

The unit vector resulted from normalizing the vector $\mathbf { w } = \langle 1,3 , - 1 \rangle$ is

Accepted Answer

A)  $\left( \frac { 1 } { \sqrt { 13 } } , \frac { 3 } { \sqrt { 13 } } , \frac { - 1 } { \sqrt { 13 } } \right)$ 
B)  $\left( \frac { 1 } { \sqrt { 11 } } , \frac { 3 } { \sqrt { 11 } } , \frac { - 1 } { \sqrt { 11 } } \right)$ 
C)  $\left( \frac { 1 } { \sqrt { 17 } } , \frac { 3 } { \sqrt { 17 } } , \frac { - 1 } { \sqrt { 17 } } \right)$ 
D)  $\left( \frac { 1 } { \sqrt { 23 } } , \frac { 3 } { \sqrt { 23 } } , \frac { - 1 } { \sqrt { 23 } } \right)$ 
E)  $\left( \frac { 1 } { \sqrt { 21 } } , \frac { 3 } { \sqrt { 21 } } , \frac { - 1 } { \sqrt { 21 } } \right)$ 
A)  $\left( \frac { 1 } { \sqrt { 13 } } , \frac { 3 } { \sqrt { 13 } } , \frac { - 1 } { \sqrt { 13 } } \right)$ 
B)  $\left( \frac { 1 } { \sqrt { 11 } } , \frac { 3 } { \sqrt { 11 } } , \frac { - 1 } { \sqrt { 11 } } \right)$ 
C)  $\left( \frac { 1 } { \sqrt { 17 } } , \frac { 3 } { \sqrt { 17 } } , \frac { - 1 } { \sqrt { 17 } } \right)$ 
D)  $\left( \frac { 1 } { \sqrt { 23 } } , \frac { 3 } { \sqrt { 23 } } , \frac { - 1 } { \sqrt { 23 } } \right)$ 
E)  $\left( \frac { 1 } { \sqrt { 21 } } , \frac { 3 } { \sqrt { 21 } } , \frac { - 1 } { \sqrt { 21 } } \right)$

Question 8

Which of the following is not true?
​

Accepted Answer

A) Vector addition is associative. 
B) Scalar addition is associative. 
C) Scalar multiplication is commutative. 
D) The quotient of two nonzero scalars is a scalar. 
E) The quotient of two nonzero vectors is a scalar. 
A) Vector addition is associative. 
B) Scalar addition is associative. 
C) Scalar multiplication is commutative. 
D) The quotient of two nonzero scalars is a scalar. 
E) The quotient of two nonzero vectors is a scalar.

Question 9

The distance between $P _ { 1 } = ( - 2,0,1 )$ and $P _ { 2 } = ( 2,4,5 )$ is

Accepted Answer

A)  $\sqrt { 47 }$ 
B)  $4 \sqrt { 3 }$ 
C) 7 
D)  $3 \sqrt { 2 }$ 
E)  $\sqrt { 19 }$ 
A)  $\sqrt { 47 }$ 
B)  $4 \sqrt { 3 }$ 
C) 7 
D)  $3 \sqrt { 2 }$ 
E)  $\sqrt { 19 }$

Question 10

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

Accepted Answer

A) 106.77 
B) 104.98 
C) 102.28 
D) 99.27 
E) 89.11 
A) 106.77 
B) 104.98 
C) 102.28 
D) 99.27 
E) 89.11

Question 11

The value of a that makes $\mathbf { v } = 6 \mathbf { i } - 4 \mathbf { j } + \mathbf { k }$ and $\mathbf { w } = a \mathbf { i } - \mathbf { j } - \mathbf { k }$ orthogonal is

Accepted Answer

A) -1 
B)  $- \frac { 1 } { 2 }$ 
C) -2 
D)  $\frac { 1 } { 2 }$ 
E) 2 
A) -1 
B)  $- \frac { 1 } { 2 }$ 
C) -2 
D)  $\frac { 1 } { 2 }$ 
E) 2

Question 12

The point of intersection of the line $x = 2 , y = 6 t + 3 , z = 3 t + 1$ and the plane $2 x - y + 3 z = 6$ is

Accepted Answer

A)  $( - 2,7,3 )$ 
B)  $( 2,7,3 )$ 
C)  $( 2 , - 7,3 )$ 
D)  $( 2,7 , - 3 )$ 
E)  $( 2 , - 7 , - 3 )$ 
A)  $( - 2,7,3 )$ 
B)  $( 2,7,3 )$ 
C)  $( 2 , - 7,3 )$ 
D)  $( 2,7 , - 3 )$ 
E)  $( 2 , - 7 , - 3 )$

Question 13

The parametric equations of the line through (1,4,-3) perpendicular to each of the lines $\frac { x + 1 } { 2 } = \frac { y + 4 } { 3 } = \frac { z - 3 } { 5 }$ and $\frac { x - 3 } { 3 } = \frac { y } { 1 } = \frac { z + 1 } { 2 }$ is

Accepted Answer

A)  $x = t + 1 , y = 11 t + 4 , z = - 7 t - 3$ 
B)  $x = t - 1 , y = 11 t - 4 , z = 7 t + 3$ 
C)  $x = t - 1 , y = 11 t - 4 , z = - 7 t - 3$ 
D)  $x = t - 1 , y = 11 t + 4 , z = - 7 t + 3$ 
E)  $x = t + 1 , y = 11 t - 4 , z = 7 t - 3$ 
A)  $x = t + 1 , y = 11 t + 4 , z = - 7 t - 3$ 
B)  $x = t - 1 , y = 11 t - 4 , z = 7 t + 3$ 
C)  $x = t - 1 , y = 11 t - 4 , z = - 7 t - 3$ 
D)  $x = t - 1 , y = 11 t + 4 , z = - 7 t + 3$ 
E)  $x = t + 1 , y = 11 t - 4 , z = 7 t - 3$

Question 14

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

Accepted Answer

A) 172.43 
B) 169.98 
C) 167.24 
D) 165.02 
E) 162.11 
A) 172.43 
B) 169.98 
C) 167.24 
D) 165.02 
E) 162.11

Question 15

The distance between the parallel planes $4 y - 3 z = 6$ and $8 y - 6 z = 27$ is

Accepted Answer

A)  $\frac { 3 } { 2 }$ 
B)  $\frac { 5 } { 3 }$ 
C)  $\frac { 5 } { 2 }$ 
D)  $\frac { 2 } { 3 }$ 
E)  $\frac { 3 } { 5 }$ 
A)  $\frac { 3 } { 2 }$ 
B)  $\frac { 5 } { 3 }$ 
C)  $\frac { 5 } { 2 }$ 
D)  $\frac { 2 } { 3 }$ 
E)  $\frac { 3 } { 5 }$

Question 16

Which of the following is not a vector?
​

Accepted Answer

A) Force 
B) The product of a scalar and a vector 
C) The area bounded by a curve 
D) Velocity 
E) Acceleration 
A) Force 
B) The product of a scalar and a vector 
C) The area bounded by a curve 
D) Velocity 
E) Acceleration

Question 17

In the three-dimensional Euclidean space, the graph of $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 9 } - \frac { z ^ { 2 } } { 16 } = - 1$ is

Accepted Answer

A) An elliptic cylinder 
B) A hyperboloid of one sheet 
C) A hyperbolic cylinder 
D) A hyperboloid of two sheets 
E) An ellipsoid 
A) An elliptic cylinder 
B) A hyperboloid of one sheet 
C) A hyperbolic cylinder 
D) A hyperboloid of two sheets 
E) An ellipsoid

Question 18

Let $P _ { 1 } = ( 3,0,2 )$ and $P _ { 2 } = ( 6,2,1 )$ v is $\overrightarrow { P _ { 1 } P _ { 2 } }$ in standard position, then v is

Accepted Answer

A)  $\langle - 3 , - 2,1 \rangle$ 
B)  $\langle 3,2 , - 1 \rangle$ 
C)  $\langle 3 , - 2,1 \rangle$ 
D)  $\langle - 3 , - 2 , - 1 \rangle$ 
E)  $\langle - 3,2,1 \rangle$ 
A)  $\langle - 3 , - 2,1 \rangle$ 
B)  $\langle 3,2 , - 1 \rangle$ 
C)  $\langle 3 , - 2,1 \rangle$ 
D)  $\langle - 3 , - 2 , - 1 \rangle$ 
E)  $\langle - 3,2,1 \rangle$

Question 19

The equation of the sphere centered at (2,1,-2) and tangent to the xy-plane is

Accepted Answer

A)  $( x + 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z + 2 ) ^ { 2 } = 2$ 
B)  $( x - 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z + 2 ) ^ { 2 } = 4$ 
C)  $( x + 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z + 2 ) ^ { 2 } = 4$ 
D)  $( x - 2 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + ( z + 2 ) ^ { 2 } = 2$ 
E)  $( x - 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z - 2 ) ^ { 2 } = 4$ 
A)  $( x + 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z + 2 ) ^ { 2 } = 2$ 
B)  $( x - 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z + 2 ) ^ { 2 } = 4$ 
C)  $( x + 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z + 2 ) ^ { 2 } = 4$ 
D)  $( x - 2 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + ( z + 2 ) ^ { 2 } = 2$ 
E)  $( x - 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z - 2 ) ^ { 2 } = 4$

Question 20

The equation of the sphere centered at (-3,2,1) through the point (4,-1,3) is

Accepted Answer

A)  $( x - 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } + ( z + 1 ) ^ { 2 } = 62$ 
B)  $( x + 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } + ( z + 1 ) ^ { 2 } = 60$ 
C)  $( x + 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } + ( z - 1 ) ^ { 2 } = 62$ 
D)  $( x + 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } + ( z - 1 ) ^ { 2 } = 62$. 
E)  $( x - 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } + ( z - 1 ) ^ { 2 } = 60$ 
A)  $( x - 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } + ( z + 1 ) ^ { 2 } = 62$ 
B)  $( x + 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } + ( z + 1 ) ^ { 2 } = 60$ 
C)  $( x + 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } + ( z - 1 ) ^ { 2 } = 62$ 
D)  $( x + 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } + ( z - 1 ) ^ { 2 } = 62$. 
E)  $( x - 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } + ( z - 1 ) ^ { 2 } = 60$

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

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

The equation of the sphere centered at (7,-1,3) and tangent to the yz-plane is

Which of the following is not true?

The equation of the plane through (1,-1,2) normal to $2 \mathbf { i } - \mathbf { j } + \mathbf { k }$ is

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

The unit vector resulted from normalizing the vector $\mathbf { w } = \langle 1,3 , - 1 \rangle$ is

Which of the following is not true?

The distance between $P _ { 1 } = ( - 2,0,1 )$ and $P _ { 2 } = ( 2,4,5 )$ is

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

The value of a that makes $\mathbf { v } = 6 \mathbf { i } - 4 \mathbf { j } + \mathbf { k }$ and $\mathbf { w } = a \mathbf { i } - \mathbf { j } - \mathbf { k }$ orthogonal is

The point of intersection of the line $x = 2 , y = 6 t + 3 , z = 3 t + 1$ and the plane $2 x - y + 3 z = 6$ is

The parametric equations of the line through (1,4,-3) perpendicular to each of the lines $\frac { x + 1 } { 2 } = \frac { y + 4 } { 3 } = \frac { z - 3 } { 5 }$ and $\frac { x - 3 } { 3 } = \frac { y } { 1 } = \frac { z + 1 } { 2 }$ is

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

The distance between the parallel planes $4 y - 3 z = 6$ and $8 y - 6 z = 27$ is

Which of the following is not a vector?

In the three-dimensional Euclidean space, the graph of $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 9 } - \frac { z ^ { 2 } } { 16 } = - 1$ is

Let $P _ { 1 } = ( 3,0,2 )$ and $P _ { 2 } = ( 6,2,1 )$ v is $\overrightarrow { P _ { 1 } P _ { 2 } }$ in standard position, then v is

The equation of the sphere centered at (2,1,-2) and tangent to the xy-plane is

The equation of the sphere centered at (-3,2,1) through the point (4,-1,3) is

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

In the three-dimensional Euclidean space, the graph of z=y24z = \frac { y ^ { 2 } } { 4 }z=4y2​ 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

The equation of the sphere centered at (7,-1,3) and tangent to the yz-plane is

Which of the following is not true? ​

The equation of the plane through (1,-1,2) normal to 2i−j+k2 \mathbf { i } - \mathbf { j } + \mathbf { k }2i−j+k is

In the three-dimensional Euclidean space, the graph of x24+y29=1\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 9 } = 14x2​+9y2​=1 is

The unit vector resulted from normalizing the vector w=⟨1,3,−1⟩\mathbf { w } = \langle 1,3 , - 1 \ranglew=⟨1,3,−1⟩ is

Which of the following is not true? ​

The distance between P1=(−2,0,1)P _ { 1 } = ( - 2,0,1 )P1​=(−2,0,1) and P2=(2,4,5)P _ { 2 } = ( 2,4,5 )P2​=(2,4,5) is

The angle, in degrees and correct to two decimal places, between 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 value of a that makes v=6i−4j+k\mathbf { v } = 6 \mathbf { i } - 4 \mathbf { j } + \mathbf { k }v=6i−4j+k and w=ai−j−k\mathbf { w } = a \mathbf { i } - \mathbf { j } - \mathbf { k }w=ai−j−k orthogonal is

The point of intersection of the line x=2,y=6t+3,z=3t+1x = 2 , y = 6 t + 3 , z = 3 t + 1x=2,y=6t+3,z=3t+1 and the plane 2x−y+3z=62 x - y + 3 z = 62x−y+3z=6 is

The angle, in degrees and correct to two decimal places, between 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 distance between the parallel planes 4y−3z=64 y - 3 z = 64y−3z=6 and 8y−6z=278 y - 6 z = 278y−6z=27 is

Which of the following is not a vector? ​

In the three-dimensional Euclidean space, the graph of x24+y29−z216=−1\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 9 } - \frac { z ^ { 2 } } { 16 } = - 14x2​+9y2​−16z2​=−1 is

Let P1=(3,0,2)P _ { 1 } = ( 3,0,2 )P1​=(3,0,2) and P2=(6,2,1)P _ { 2 } = ( 6,2,1 )P2​=(6,2,1) v is P1P2→\overrightarrow { P _ { 1 } P _ { 2 } }P1​P2​​ in standard position, then v is

The equation of the sphere centered at (2,1,-2) and tangent to the xy-plane is

The equation of the sphere centered at (-3,2,1) through the point (4,-1,3) is

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

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

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

Which of the following is not true?

The equation of the plane through (1,-1,2) normal to $2 \mathbf { i } - \mathbf { j } + \mathbf { k }$ is

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

The unit vector resulted from normalizing the vector $\mathbf { w } = \langle 1,3 , - 1 \rangle$ is

Which of the following is not true?

The distance between $P _ { 1 } = ( - 2,0,1 )$ and $P _ { 2 } = ( 2,4,5 )$ is

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

The value of a that makes $\mathbf { v } = 6 \mathbf { i } - 4 \mathbf { j } + \mathbf { k }$ and $\mathbf { w } = a \mathbf { i } - \mathbf { j } - \mathbf { k }$ orthogonal is

The point of intersection of the line $x = 2 , y = 6 t + 3 , z = 3 t + 1$ and the plane $2 x - y + 3 z = 6$ is

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

The distance between the parallel planes $4 y - 3 z = 6$ and $8 y - 6 z = 27$ is

Which of the following is not a vector?

In the three-dimensional Euclidean space, the graph of $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 9 } - \frac { z ^ { 2 } } { 16 } = - 1$ is

Let $P _ { 1 } = ( 3,0,2 )$ and $P _ { 2 } = ( 6,2,1 )$ v is $\overrightarrow { P _ { 1 } P _ { 2 } }$ in standard position, then v is