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

Question 1

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

A) $\sqrt { 13 }$
B) $\sqrt { 15 }$
C) $\sqrt { 17 }$
D) $\sqrt { 19 }$
E) $\sqrt { 21 }$

Answer 1

$\sqrt { 17 }$

Question 2

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

A) $\sqrt { 19 }$
B) $\sqrt { 21 }$
C) $\sqrt { 23 }$
D) $\sqrt { 29 }$
E) $\sqrt { 33 }$

Answer 2

$\sqrt { 33 }$

Question 3

The distance between $P _ { 1 } = ( 0,4,1 )$ and $P _ { 2 } = ( 1,8,2 )$ is

A) $\sqrt { 17 }$
B) $3 \sqrt { 2 }$
C) $\sqrt { 19 }$
D) $\sqrt { 21 }$
E) $\sqrt { 23 }$

Answer 3

$3 \sqrt { 2 }$

Question 4

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

A) $\sqrt { 13 }$
B) $\sqrt { 11 }$
C) $\sqrt { 7 }$
D) $\sqrt { 5 }$
E) $\sqrt { 3 }$

Question 5

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

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

Question 6

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

A) $\sqrt { 26 }$
B) $3 \sqrt { 3 }$
C) $\sqrt { 29 }$
D) $4 \sqrt { 2 }$
E) $\sqrt { 43 }$

Question 7

The distance between $P _ { 1 } = ( 5,6 , - 1 )$ and $P _ { 2 } = ( 3,8,2 )$ is

A) $\sqrt { 15 }$
B)4
C) $\sqrt { 17 }$
D) $3 \sqrt { 2 }$
E) $\sqrt { 19 }$

Question 8

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

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

Question 9

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

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

Question 10

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

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

Question 11

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

A) $( - 2,0,0 ) ; 2$
B) $( 2,0,0 ) ; 2$
C) $( - 2,0,0 ) ; 1$
D) $( 2,0,0 ) ; 4$
E) $( - 2,0,0 ) ; 4$

Question 12

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

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 13

The center and radius of the sphere $3 x ^ { 2 } + 3 y ^ { 2 } + 3 z ^ { 2 } + 6 x - z - 3 = 0$ are

A) $\left( - 1,0 , - \frac { 1 } { 6 } \right) ; 2$
B) $\left( - 1,0 , \frac { 1 } { 6 } \right) ; 7$
C) $( - 1,0 , - 6 ) ; 5$
D) $\left( - 1,0 , \frac { 1 } { 6 } \right) ; \frac { \sqrt { 73 } } { 6 }$
E) $\left( 1,0 , \frac { 1 } { 6 } \right) ; \sqrt { 13 }$

Question 14

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

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 15

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

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 16

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

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 17

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

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) $( x - 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } + ( z - 1 ) ^ { 2 } = 60$

Question 18

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

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 19

The equation of the sphere centered at (1,5,4) and tangent to the xz-plane is

A) $( x - 1 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z + 4 ) ^ { 2 } = 16$
B) $( x + 1 ) ^ { 2 } + ( y - 5 ) ^ { 2 } + ( z + 4 ) ^ { 2 } = 25$
C) $( x + 1 ) ^ { 2 } + ( y - 5 ) ^ { 2 } + ( z - 4 ) ^ { 2 } = 25$
D) $( x - 1 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 4 ) ^ { 2 } = 16$
E) $( x - 1 ) ^ { 2 } + ( y - 5 ) ^ { 2 } + ( z - 4 ) ^ { 2 } = 25$

Question 20

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

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 21

Which of the following is not true?

A) $\mathbf { v } + \mathbf { w } = \mathbf { w } + \mathbf { v }$
B) $\mathbf { v } + ( - \mathbf { w } ) = \mathbf { v } - \mathbf { w }$
C) $\frac { \mathbf { v } } { \mathbf { v } } = 1$ for a nonzero vector v
D)v + 0 = v
E)The product of a vector and scalar is a vector.

Question 22

Which of the following is not a scalar?
​

A)Distance
B)Volume
C)Velocity
D)Length
E)Speed

Question 23

Which of the following is true?
​

A)Area is a vector.
B)Volume is a vector.
C)Velocity is a scalar.
D)Acceleration is a vector.
E)Force is a scalar.

Question 24

Which of the following is not true?
​

A)The product of two scalars is a scalar.
B)The sum of two vectors is a vector.
C)The difference of two vectors is a vector.
D)Acceleration is a vector.
E)The product of a vector and a scalar is a scalar.

Question 25

Which of the following is not true?

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 26

Which of the following is not true?
​

A)The sum of two scalars is a scalar
B)The sum of two vectors is a vector.
C)The difference of two vectors is a vector.
D)The difference of two scalars is a scalar.
E)The sum of a scalar and a vector is a vector.

Question 27

Which of the following is not necessarily true?
​

A)The product of two scalars is a scalar.
B)The product of two vectors is a vector.
C)The sum of two vectors is a vector.
D)The difference of two scalars is a scalar.
E)The sum two scalars is a scalar.

Question 28

Which of the following is not a vector?
​

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

Question 29

Which of the following is not true?

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 30

Which of the following is not true?
​

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 31

Which of the following is not true?
​

A)Vector addition is commutative.
B)Scalar addition is commutative.
C)Scalar multiplication is commutative.
D)The product of a scalar and a vector is a scalar.
E)The product of a vector and a scalar is a vector.

Question 32

Which of the following is not true?
​

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

Question 33

Which of the following is not true?
​

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

Question 34

Which of the following is not true?
​

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

Question 35

Which of the following is not true?
​

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 36

Which of the following is not true?
​

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

Question 37

Which of the following is not true?
​

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.

Question 38

Which of the following is not true?
​

A)Vector addition is commutative.
B)Scalar addition is commutative.
C)The sum of two vectors is a vector.
D)The sum of two scalars is a scalar.
E)The area of a region bounded by a curve is a vector.

Question 39

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

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 40

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

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 41

Let $P _ { 1 } = ( 1,3,5 )$ and $P _ { 2 } = ( - 2,4 , - 6 )$ v is $\overrightarrow { P _ { 1 } P _ { 2 } }$ in standard position, then v is

A) $\langle 3,1 , - 11 \rangle$
B) $\langle 3 , - 1 , - 11 \rangle$
C) $\langle - 3 , - 1 , - 11 \rangle$
D) $\langle - 3 , - 1,11 \rangle$
E) $\langle - 3,1 , - 11 \rangle$

Question 42

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

A) $\langle - 5 , - 3 , - 1 \rangle$
B) $\langle 5 , - 3 , - 1 \rangle$
C) $\langle 5 , - 3,1 \rangle$
D) $\langle - 5,3 , - 1 \rangle$
E) $\langle - 5 , - 3,1 \rangle$

Question 43

Let $P _ { 1 } = ( 0,1 , - 4 )$ and $P _ { 2 } = ( - 2,8,0 )$ If v is $\overrightarrow { P _ { 1 } P _ { 2 } }$ in standard position, then v is

A) $\langle - 2,7 , - 4 \rangle$
B) $\langle - 2,7,4 \rangle$
C) $\langle - 2,7 , - 4 \rangle$ .
D) $\langle 2 , - 7,4 \rangle$
E) $\langle 2,7 , - 4 \rangle$ .

Question 44

Let $P _ { 1 } = ( 3 , - 2,5 )$ and $P _ { 2 } = ( 10,4,3 )$ If v is $\overrightarrow { P _ { 1 } P _ { 2 } }$ in standard position, then v is

A) $\langle 7 , - 6 , - 2 \rangle$
B) $\langle 7 , - 6,2 \rangle$
C) $\langle 7,6 , - 2 \rangle$
D) $\langle 7,6,2 \rangle$
E) $\langle - 7,6 , - 2 \rangle$

Question 45

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

A) $\langle - 25,2 , - 44 \rangle$
B) $\langle - 25,2,44 \rangle$
C) $\langle 25 , - 2 , - 44 \rangle$
D) $\langle - 25 , - 2 , - 44 \rangle$
E) $\langle - 25 , - 2,44 \rangle$

Question 46

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

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 47

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

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 48

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

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 49

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

A) $\sqrt { 53 }$
B) $\sqrt { 54 }$
C) $\sqrt { 55 }$
D) $\sqrt { 56 }$
E) $\sqrt { 57 }$

Question 50

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

A) $\sqrt { 23 } - \sqrt { 14 }$
B) $\sqrt { 23 } + \sqrt { 14 }$
C) $\sqrt { 23 } - \sqrt { 17 }$
D) $\sqrt { 21 } + \sqrt { 14 }$
E) $\sqrt { 21 } - \sqrt { 14 }$

Question 51

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

A) $\sqrt { 38 } - \sqrt { 73 }$
B) $\sqrt { 38 } - \sqrt { 71 }$
C) $\sqrt{73}-\sqrt{38}$

D) $\sqrt { 38 } + \sqrt { 71 }$
E) $\sqrt { 38 } + \sqrt { 73 }$

Question 52

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

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 53

Let $\mathbf { u } = \langle 1,1 , - 6 \rangle , \mathbf { v } = \langle 3,0,8 \rangle$ , and $\mathbf { w } = \langle - 3 , - 1,2 \rangle$ Then $\| \mathbf { u } \| + \| \mathbf { v } \| - \| \mathbf { w } \|$ is

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

Question 54

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

A) $\sqrt { 119 } - \sqrt { 14 }$
B) $\sqrt { 203 } - \sqrt { 14 }$
C) $\sqrt { 201 } - \sqrt { 14 }$
D) $\sqrt { 201 } - \sqrt { 13 }$
E) $\sqrt { 203 } - \sqrt { 13 }$

Question 55

The unit vector resulted from normalizing the vector $\mathbf { w } = \langle 2 , - 1,4 \rangle$ is

A) $\left( \frac { 2 } { \sqrt { 19 } } , \frac { - 1 } { \sqrt { 19 } } , \frac { 4 } { \sqrt { 19 } } \right)$
B) $\left( \frac { 2 } { \sqrt { 21 } } , \frac { - 1 } { \sqrt { 21 } } , \frac { 4 } { \sqrt { 21 } } \right)$
C) $\left( \frac { 2 } { 5 } , \frac { - 1 } { 5 } , \frac { 4 } { 5 } \right)$
D) $\left( \frac { 2 } { \sqrt { 23 } } , \frac { - 1 } { \sqrt { 23 } } , \frac { 4 } { \sqrt { 23 } } \right)$
E) $\left( \frac { 2 } { \sqrt { 29 } } , \frac { - 1 } { \sqrt { 29 } } , \frac { 4 } { \sqrt { 29 } } \right)$

Question 56

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

A) $\frac { 2 } { \sqrt { 19 } } \mathbf { i } - \frac { 3 } { \sqrt { 19 } } \mathbf { j } - \frac { 4 } { \sqrt { 19 } } \mathbf { k }$
B) $\frac { 2 } { \sqrt { 21 } } \mathbf { i } - \frac { 3 } { \sqrt { 21 } } \mathrm { j } - \frac { 4 } { \sqrt { 21 } } \mathbf { k }$
C) $\frac { 2 } { \sqrt { 23 } } \mathbf { i } - \frac { 3 } { \sqrt { 23 } } \mathbf { j } - \frac { 4 } { \sqrt { 23 } } \mathbf { k }$
D) $\frac { 2 } { 5 } \mathrm { i } - \frac { 3 } { 5 } \mathrm { j } - \frac { 4 } { 5 } \mathrm { k }$
E) $\frac { 2 } { \sqrt { 29 } } \mathbf { i } - \frac { 3 } { \sqrt { 29 } } \mathbf { j } - \frac { 4 } { \sqrt { 29 } } \mathbf { k }$

Question 57

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

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 58

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

A) $\frac { 1 } { 3 } \mathbf { i } + \frac { 1 } { 3 } \mathbf { j } + \frac { 1 } { 3 } \mathbf { k }$
B) $\frac { 1 } { \sqrt { 7 } } i + \frac { 1 } { \sqrt { 7 } } j + \frac { 1 } { \sqrt { 7 } } \mathbf { k }$
C) $\frac { 1 } { \sqrt { 5 } } \mathbf { i } + \frac { 1 } { \sqrt { 5 } } \mathbf { j } + \frac { 1 } { \sqrt { 5 } } \mathbf { k }$
D) $\frac { 1 } { \sqrt { 3 } } \mathbf { i } + \frac { 1 } { \sqrt { 3 } } \mathbf { j } + \frac { 1 } { \sqrt { 3 } } \mathbf { k }$
E) $\frac { 1 } { \sqrt { 2 } } \mathbf { i } + \frac { 1 } { \sqrt { 2 } } \mathbf { j } + \frac { 1 } { \sqrt { 2 } } \mathbf { k }$

Question 59

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

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

Question 60

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

A)7
B)11
C)13
D)-11
E)-7

Question 61

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

A)2
B)5
C)6
D)-2
E)-6

Question 62

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

A)2
B)5
C)6
D)-2
E)-6

Question 63

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

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

Question 64

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

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

Question 65

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

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

Question 66

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

A)3
B) $\frac { 7 } { 4 }$
C)3.
D) $- \frac { 7 } { 4 }$
E) $- \frac { 9 } { 4 }$

Question 67

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

A)25.24
B)24.38
C)22.21
D)21.68
E)20.77

Question 68

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

A)113.34
B)112.54
C)111.98
D)110.32
E)109.11

Question 69

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

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

Question 70

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

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

Question 71

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

A)114.77
B)111.96
C)107.67
D)105.67
E)104.11

Question 72

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

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

Question 73

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

A)138.77
B)122.68
C)111.98
D)109.28
E)99.11

Question 74

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

A) $- \frac { 2 } { 11 } \mathbf { i } - \frac { 1 } { 11 } \mathbf { j } + \frac { 3 } { 11 } \mathbf { k }$
B) $- \frac { 2 } { 7 } \mathbf { i } + \frac { 1 } { 7 } \mathbf { j } + \frac { 3 } { 7 } \mathbf { k }$
C) $- \frac { 2 } { 7 } \mathrm { i } - \frac { 1 } { 7 } \mathrm { j } + \frac { 3 } { 7 } \mathrm { k }$
D) $\frac { 2 } { 7 } \mathrm { i } - \frac { 1 } { 7 } \mathrm { j } + \frac { 3 } { 7 } \mathrm { k }$
E) $\frac { 2 } { 7 } i + \frac { 1 } { 7 } j + \frac { 3 } { 7 } k$

Question 75

The vector projection of $\mathbf { v } = 0$ onto $\mathbf { w } = 2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k }$ is

A) $2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k }$
B) $0 \mathbf { i } + 0 \mathbf { j } + 0 \mathbf { k }$
C) $\frac { 2 } { 14 } i + \frac { 1 } { 14 } \mathbf { j } - \frac { 3 } { 14 } \mathbf { k }$
D) $- \frac { 2 } { 14 } \mathbf { i } - \frac { 1 } { 14 } \mathbf { j } + \frac { 3 } { 14 } \mathbf { k }$
E)Undefined

Question 76

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

A) $- \frac { 38 } { 13 } \mathbf { i } + \frac { 19 } { 26 } j - \frac { 57 } { 26 } k$
B) $\frac { 38 } { 13 } \mathbf { i } - \frac { 19 } { 26 } \mathbf { j } - \frac { 57 } { 26 } \mathbf { k }$
C) $- \frac { 38 } { 13 } i - \frac { 19 } { 26 } j + \frac { 57 } { 26 } k$
D) $- \frac { 38 } { 13 } \mathbf { i } + \frac { 19 } { 26 } \mathbf { j } + \frac { 57 } { 26 } \mathbf { k }$
E) $\frac { 38 } { 13 } \mathbf { i } - \frac { 19 } { 26 } \mathbf { j } + \frac { 57 } { 26 } \mathbf { k }$

Question 77

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

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 78

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

A) $- \mathbf { i } + 5 \mathbf { j } - 7 \mathbf { k }$
B) $- \mathbf { i } - 5 \mathbf { j } + 7 \mathbf { k }$
C) $- \mathbf { i } - 5 \mathbf { j } - 7 \mathbf { k }$
D) $\mathbf { i } + 5 \mathbf { j } - 7 \mathbf { k }$
E) $\mathbf { i } - 5 \mathbf { j } - 7 \mathbf { k }$

Question 79

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

A) $- 2 \mathbf { i } - \mathbf { j } + 5 \mathbf { k }$
B) $2 \mathbf { i } - \mathbf { j } - 5 \mathbf { k }$
C) $- 2 \mathbf { i } + \mathbf { j } + 5 \mathbf { k }$
D) $2 \mathbf { i } + \mathbf { j } + 5 \mathbf { k }$
E) $2 \mathbf { i } - \mathbf { j } + 5 \mathbf { k }$

Question 80

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

A) $5 \mathbf { i } + 13 \mathbf { j } + 11 \mathbf { k }$
B) $- 5 \mathbf { i } + 13 \mathbf { j } + 11 \mathbf { k }$
C) $- 5 \mathbf { i } + 13 \mathbf { j } - 11 \mathbf { k }$
D) $5 \mathbf { i } + 13 \mathbf { j } - 11 \mathbf { k }$
E) $5 \mathbf { i } - 13 \mathbf { j } + 11 \mathbf { k }$