Question 1

The center and radius of the sphere whose endpoints of a diameter (-2, 0, 4) and (2,6,8) are

Accepted Answer

A)  $( 0 , - 3 , - 6 ) ; 4$ 
B)  $( 0 , - 3,6 ) ; \sqrt { 17 }$ 
C)  $( 0,3 , - 6 ) ; \sqrt { 17 }$ 
D)  $( 0,3,6 ) ; \sqrt { 17 }$ 
E)  $( 0,3,6 ) ; 4$ 
A)  $( 0 , - 3 , - 6 ) ; 4$ 
B)  $( 0 , - 3,6 ) ; \sqrt { 17 }$ 
C)  $( 0,3 , - 6 ) ; \sqrt { 17 }$ 
D)  $( 0,3,6 ) ; \sqrt { 17 }$ 
E)  $( 0,3,6 ) ; 4$

Question 2

Which of the following is not true?

Accepted Answer

A)  $\mathbf { v } + 0 = \mathbf { v }$ 
B)  $\mathbf { u } - \mathbf { v } = - \mathbf { v } + \mathbf { u }$ 
C)  $\frac { 2 \mathbf { v } } { \mathbf { v } } = 2$ for a nonzero vector 
D)  $- 3 ( \mathbf { u } - \mathbf { v } ) = - 3 \mathbf { u } + 3 \mathbf { v }$ 
E)  $\frac { 2 \mathbf { v } } { 3 } = \frac { 2 } { 3 } \mathbf { v }$ 
A)  $\mathbf { v } + 0 = \mathbf { v }$ 
B)  $\mathbf { u } - \mathbf { v } = - \mathbf { v } + \mathbf { u }$ 
C)  $\frac { 2 \mathbf { v } } { \mathbf { v } } = 2$ for a nonzero vector 
D)  $- 3 ( \mathbf { u } - \mathbf { v } ) = - 3 \mathbf { u } + 3 \mathbf { v }$ 
E)  $\frac { 2 \mathbf { v } } { 3 } = \frac { 2 } { 3 } \mathbf { v }$

Question 3

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

Accepted Answer

A)  $\langle - 1,1 , - 10 \rangle$ 
B)  $\langle - 1 , - 1 , - 10 \rangle$ 
C)  $\langle 1 , - 1 , - 10 \rangle$ 
D)  $\langle 1 , - 1,10 \rangle$ 
E)  $\langle - 1 , - 1,10 \rangle$ 
A)  $\langle - 1,1 , - 10 \rangle$ 
B)  $\langle - 1 , - 1 , - 10 \rangle$ 
C)  $\langle 1 , - 1 , - 10 \rangle$ 
D)  $\langle 1 , - 1,10 \rangle$ 
E)  $\langle - 1 , - 1,10 \rangle$

Question 4

Which of the following is not true?

Accepted Answer

A) The sum of two scalars is a scalar. 
B) The sum of two vectors is a vector. 
C) v + 0 = v 
D)  $\mathbf { v } - \mathbf { w } = \mathbf { v } + ( - \mathbf { w } )$ 
E) Force is a vector. 
A) The sum of two scalars is a scalar. 
B) The sum of two vectors is a vector. 
C) v + 0 = v 
D)  $\mathbf { v } - \mathbf { w } = \mathbf { v } + ( - \mathbf { w } )$ 
E) Force is a vector.

Question 5

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

Accepted Answer

A) 154.76 
B) 157.95 
C) 159.43 
D) 162.54 
E) 165.11 
A) 154.76 
B) 157.95 
C) 159.43 
D) 162.54 
E) 165.11

Question 6

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 7

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

Accepted Answer

A) 6 
B) 4 
C) -4 
D) -2 
E) -8 
A) 6 
B) 4 
C) -4 
D) -2 
E) -8

Question 8

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

Accepted Answer

A)  $\frac { 5 } { 3 } i - \frac { 5 } { 3 } j + \frac { 5 } { 3 } \mathbf { k }$ 
B)  $\frac { 5 } { 3 } i - \frac { 5 } { 3 } j - \frac { 5 } { 3 } k$ 
C)  $- \frac { 5 } { 3 } i + \frac { 5 } { 3 } j + \frac { 5 } { 3 } k$ 
D)  $- \frac { 5 } { 3 } i - \frac { 5 } { 3 } j + \frac { 5 } { 3 } k$ 
E)  $\frac { 5 } { 3 } i + \frac { 5 } { 3 } j - \frac { 5 } { 3 } \mathbf { k }$ 
A)  $\frac { 5 } { 3 } i - \frac { 5 } { 3 } j + \frac { 5 } { 3 } \mathbf { k }$ 
B)  $\frac { 5 } { 3 } i - \frac { 5 } { 3 } j - \frac { 5 } { 3 } k$ 
C)  $- \frac { 5 } { 3 } i + \frac { 5 } { 3 } j + \frac { 5 } { 3 } k$ 
D)  $- \frac { 5 } { 3 } i - \frac { 5 } { 3 } j + \frac { 5 } { 3 } k$ 
E)  $\frac { 5 } { 3 } i + \frac { 5 } { 3 } j - \frac { 5 } { 3 } \mathbf { k }$

Question 9

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

Accepted Answer

A)  $\langle 16,4,4 \rangle$ 
B)  $\langle 16,4 , - 4 \rangle$ 
C)  $\langle 16 , - 4,4 \rangle$ 
D)  $\langle 16 , - 4 , - 4 \rangle$ 
E)  $\langle - 16,4,4 \rangle$ 
A)  $\langle 16,4,4 \rangle$ 
B)  $\langle 16,4 , - 4 \rangle$ 
C)  $\langle 16 , - 4,4 \rangle$ 
D)  $\langle 16 , - 4 , - 4 \rangle$ 
E)  $\langle - 16,4,4 \rangle$

Question 10

The parametric equations of the line through (3,2,1) in the direction of $3 \mathbf { i } - \mathbf { j } + \mathbf { k }$ is

Accepted Answer

A)  $x = 3 t + 3 , y = t + 2 , z = t + 1$ 
B)  $x = 3 t + 3 , y = - t + 2 , z = t + 1$ 
C)  $x = 3 t - 3 , y = - t + 2 , z = t + 1$ 
D)  $x = 3 t + 3 , y = - t + 2 , z = t - 1$ 
E)  $x = 3 t - 3 , y = t + 2 , z = t - 1$ 
A)  $x = 3 t + 3 , y = t + 2 , z = t + 1$ 
B)  $x = 3 t + 3 , y = - t + 2 , z = t + 1$ 
C)  $x = 3 t - 3 , y = - t + 2 , z = t + 1$ 
D)  $x = 3 t + 3 , y = - t + 2 , z = t - 1$ 
E)  $x = 3 t - 3 , y = t + 2 , z = t - 1$

Question 11

The center and radius of the sphere $2 x ^ { 2 } + 2 y ^ { 2 } + 2 z ^ { 2 } - 8 x + 5 z + 1 = 0$ are

Accepted Answer

A)  $\left( 2,0 , - \frac { 5 } { 4 } \right) ; \frac { 9 } { 4 }$ 
B)  $\left( 2,0 , \frac { 5 } { 4 } \right) ; \frac { 9 } { 4 }$ 
C)  $\left( - 2,0 , - \frac { 5 } { 4 } \right) ; \frac { 4 } { 9 }$ 
D)  $\left( 2,0 , - \frac { 5 } { 4 } \right) ; \frac { 4 } { 9 }$ 
E)  $\left( 2,0 , \frac { 5 } { 4 } \right) ; \frac { 4 } { 9 }$ 
A)  $\left( 2,0 , - \frac { 5 } { 4 } \right) ; \frac { 9 } { 4 }$ 
B)  $\left( 2,0 , \frac { 5 } { 4 } \right) ; \frac { 9 } { 4 }$ 
C)  $\left( - 2,0 , - \frac { 5 } { 4 } \right) ; \frac { 4 } { 9 }$ 
D)  $\left( 2,0 , - \frac { 5 } { 4 } \right) ; \frac { 4 } { 9 }$ 
E)  $\left( 2,0 , \frac { 5 } { 4 } \right) ; \frac { 4 } { 9 }$

Question 12

The cross product of $\mathbf { v } = \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k }$ and $\mathbf { w } = 3 \mathbf { i } + \mathbf { j } - 3 \mathbf { k }$ is

Accepted Answer

A)  $- 5 \mathbf { i } + 15 \mathbf { j } + 10 \mathbf { k }$ 
B)  $5 \mathbf { i } - 15 \mathbf { j } + 10 \mathbf { k }$ 
C)  $5 \mathbf { i } - 15 \mathbf { j } - 10 \mathbf { k }$ 
D)  $5 \mathbf { i } + 15 \mathbf { j } + 10 \mathbf { k }$ 
E)  $5 \mathbf { i } + 15 \mathbf { j } - 10 \mathbf { k }$ 
A)  $- 5 \mathbf { i } + 15 \mathbf { j } + 10 \mathbf { k }$ 
B)  $5 \mathbf { i } - 15 \mathbf { j } + 10 \mathbf { k }$ 
C)  $5 \mathbf { i } - 15 \mathbf { j } - 10 \mathbf { k }$ 
D)  $5 \mathbf { i } + 15 \mathbf { j } + 10 \mathbf { k }$ 
E)  $5 \mathbf { i } + 15 \mathbf { j } - 10 \mathbf { k }$

Question 13

The center and radius of the sphere whose endpoints of a diameter (1,3,6) and (-3,1,4) are

Accepted Answer

A)  $( - 1 , - 2,5 ) ; \sqrt { 7 }$ 
B)  $( - 1,2 , - 5 ) ; \sqrt { 6 }$ 
C)  $( 1 , - 2 , - 5 ) ; \sqrt { 6 }$ 
D)  $( - 1,2,5 ) ; \sqrt { 7 }$ 
E)  $( - 1 , - 2,5 ) ; \sqrt { 6 }$ 
A)  $( - 1 , - 2,5 ) ; \sqrt { 7 }$ 
B)  $( - 1,2 , - 5 ) ; \sqrt { 6 }$ 
C)  $( 1 , - 2 , - 5 ) ; \sqrt { 6 }$ 
D)  $( - 1,2,5 ) ; \sqrt { 7 }$ 
E)  $( - 1 , - 2,5 ) ; \sqrt { 6 }$

Question 14

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

Accepted Answer

A)  $3 \sqrt { 77 }$ 
B)  $3 \sqrt { 67 }$ 
C)  $3 \sqrt { 57 }$ 
D)  $3 \sqrt { 58 }$ 
E)  $3 \sqrt { 59 }$ 
A)  $3 \sqrt { 77 }$ 
B)  $3 \sqrt { 67 }$ 
C)  $3 \sqrt { 57 }$ 
D)  $3 \sqrt { 58 }$ 
E)  $3 \sqrt { 59 }$

Question 15

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

Accepted Answer

A)  $\sqrt { 38 } - \sqrt { 73 } - 2 \sqrt { 14 }$ 
B)  $\sqrt { 38 } + \sqrt { 73 } + 2 \sqrt { 14 }$ 
C)  $\sqrt { 38 } - \sqrt { 73 } + 2 \sqrt { 14 }$ 
D)  $- \sqrt { 38 } + \sqrt { 73 } - 2 \sqrt { 14 }$ 
E)  $\sqrt { 38 } + \sqrt { 73 } - 2 \sqrt { 14 }$ 
A)  $\sqrt { 38 } - \sqrt { 73 } - 2 \sqrt { 14 }$ 
B)  $\sqrt { 38 } + \sqrt { 73 } + 2 \sqrt { 14 }$ 
C)  $\sqrt { 38 } - \sqrt { 73 } + 2 \sqrt { 14 }$ 
D)  $- \sqrt { 38 } + \sqrt { 73 } - 2 \sqrt { 14 }$ 
E)  $\sqrt { 38 } + \sqrt { 73 } - 2 \sqrt { 14 }$

Question 16

The distance between $P _ { 1 } = ( - 2 , - 3,0 )$ and $P _ { 2 } = ( - 6,7,1 )$ is

Accepted Answer

A)  $3 \sqrt { 13 }$ 
B)  $\sqrt { 87 }$ 
C)  $\sqrt { 67 }$ 
D)  $\sqrt { 39 }$ 
E)  $\sqrt { 41 }$ 
A)  $3 \sqrt { 13 }$ 
B)  $\sqrt { 87 }$ 
C)  $\sqrt { 67 }$ 
D)  $\sqrt { 39 }$ 
E)  $\sqrt { 41 }$

Question 17

The center and radius of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 z = 0$ are

Accepted Answer

A)  $( 0,0 , - 2 ) ; 2$ 
B)  $( 0,0 , - 2 ) ; 2$. 
C)  $( 0,0 , - 2 ) ; 1$ 
D)  $( 0,0,2 ) ; 1$ 
E)  $( 0,0,2 ) ; 2$ 
A)  $( 0,0 , - 2 ) ; 2$ 
B)  $( 0,0 , - 2 ) ; 2$. 
C)  $( 0,0 , - 2 ) ; 1$ 
D)  $( 0,0,2 ) ; 1$ 
E)  $( 0,0,2 ) ; 2$

Question 18

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

Accepted Answer

A) 1 
B) 2 
C) 3 
D) -1 
E) -2 
A) 1 
B) 2 
C) 3 
D) -1 
E) -2

Question 19

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

Accepted Answer

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

Question 20

Which of the following is not true?
​

Accepted Answer

A) Vector addition is commutative. 
B) Scalar addition is commutative. 
C) The distance between two points is a scalar. 
D) The sum of two vectors is a scalar. 
E) The sum of two scalars is a scalar. 
A) Vector addition is commutative. 
B) Scalar addition is commutative. 
C) The distance between two points is a scalar. 
D) The sum of two vectors is a scalar. 
E) The sum of two scalars is a scalar.

The center and radius of the sphere whose endpoints of a diameter (-2, 0, 4) and (2,6,8) are

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 not true?

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

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

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

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

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

The parametric equations of the line through (3,2,1) in the direction of $3 \mathbf { i } - \mathbf { j } + \mathbf { k }$ is

The center and radius of the sphere $2 x ^ { 2 } + 2 y ^ { 2 } + 2 z ^ { 2 } - 8 x + 5 z + 1 = 0$ are

The cross product of $\mathbf { v } = \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k }$ and $\mathbf { w } = 3 \mathbf { i } + \mathbf { j } - 3 \mathbf { k }$ is

The center and radius of the sphere whose endpoints of a diameter (1,3,6) and (-3,1,4) are

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

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

The distance between $P _ { 1 } = ( - 2 , - 3,0 )$ and $P _ { 2 } = ( - 6,7,1 )$ is

The center and radius of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 z = 0$ are

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

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

Which of the following is not true?

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

The center and radius of the sphere whose endpoints of a diameter (-2, 0, 4) and (2,6,8) are

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 not true?

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

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

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

The vector projection of v=i−3j+k\mathbf { v } = \mathbf { i } - 3 \mathbf { j } + \mathbf { k }v=i−3j+k onto w=i−j+k\mathbf { w } = \mathbf { i } - \mathbf { j } + \mathbf { k }w=i−j+k 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 parametric equations of the line through (3,2,1) in the direction of 3i−j+k3 \mathbf { i } - \mathbf { j } + \mathbf { k }3i−j+k is

The center and radius of the sphere 2x2+2y2+2z2−8x+5z+1=02 x ^ { 2 } + 2 y ^ { 2 } + 2 z ^ { 2 } - 8 x + 5 z + 1 = 02x2+2y2+2z2−8x+5z+1=0 are

The cross product of 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

The center and radius of the sphere whose endpoints of a diameter (1,3,6) and (-3,1,4) are

The area of the parallelogram PQRS with vertices P=(0,1,1),Q=(2,0,−4)P = ( 0,1,1 ) , Q = ( 2,0 , - 4 )P=(0,1,1),Q=(2,0,−4) and R=(−3,−2,1)R = ( - 3 , - 2,1 )R=(−3,−2,1) 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

The distance between P1=(−2,−3,0)P _ { 1 } = ( - 2 , - 3,0 )P1​=(−2,−3,0) and P2=(−6,7,1)P _ { 2 } = ( - 6,7,1 )P2​=(−6,7,1) is

The center and radius of the sphere x2+y2+z2−4z=0x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 z = 0x2+y2+z2−4z=0 are

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

In the three-dimensional Euclidean space, the graph of x29+y24−z2=1\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } - z ^ { 2 } = 19x2​+4y2​−z2=1 is

Which of the following is not true? ​

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

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

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

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

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

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

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

The parametric equations of the line through (3,2,1) in the direction of $3 \mathbf { i } - \mathbf { j } + \mathbf { k }$ is

The center and radius of the sphere $2 x ^ { 2 } + 2 y ^ { 2 } + 2 z ^ { 2 } - 8 x + 5 z + 1 = 0$ are

The cross product of $\mathbf { v } = \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k }$ and $\mathbf { w } = 3 \mathbf { i } + \mathbf { j } - 3 \mathbf { k }$ is

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

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

The distance between $P _ { 1 } = ( - 2 , - 3,0 )$ and $P _ { 2 } = ( - 6,7,1 )$ is

The center and radius of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 z = 0$ are

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

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

Which of the following is not true?