Deck 10: Vectors

Question 1

What is the distance between the given pair of points: $( 1 , - 3 ) \text { and } ( - 2 , - 1 )$ ?

Answer 1

$\sqrt { 13 }$

Question 2

What is the distance between the given pair of points: $( 1,2 , - 1 ) \text { and } ( 4,2,2 )$ ?

Answer 2

$3 \sqrt { 2 }$

Question 3

What is the distance between the given pair of points: $( - 1 , - 2,1 ) \text { and } ( - 2,2,0 )$ ?

A) $\sqrt { 10 }$
B) $8$
C) $18$
D) $3 \sqrt { 2 }$

Answer 3

$3 \sqrt { 2 }$

Question 4

Find the equation of the sphere with center $( 1,2,1 )$ and radius 2.

Question 5

Find the equation of the sphere with center $( - 1 , - 2,4 )$ and radius $\sqrt { 3 }$

Question 6

Find the equation of the sphere with center $( 1 , - 2,3 )$ and radius 4.

A) $( x - 1 ) ^ { 2 } + ( y + 2 ) ^ { 2 } + ( z - 3 ) ^ { 2 } = 16$
B) $( x + 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } + ( z + 3 ) ^ { 2 } = 16$
C) $( x - 1 ) ^ { 2 } + ( y + 2 ) ^ { 2 } + ( z - 3 ) ^ { 2 } = 4$
D) $( x - 1 ) ^ { 2 } + ( y + 2 ) ^ { 2 } + ( z - 3 ) ^ { 2 } = 2$

Question 7

Find the equation of the sphere with the center $( 1,2,3 )$ containing the origin.

Question 8

Find the equation of the sphere with the center $( - 1,2 , - 3 )$ tangent to the $y z -$ plane.

Question 9

Find the equation of the sphere with the center $( - 2,3,4 )$ tangent to the $y z -$ plane.

A) $( x + 2 ) ^ { 2 } + ( y - 3 ) ^ { 2 } + ( z - 4 ) ^ { 2 } = 9$
B) $( x + 2 ) ^ { 2 } + ( y - 3 ) ^ { 2 } + ( z - 4 ) ^ { 2 } = 3$
C) $( x + 2 ) ^ { 2 } + ( y - 3 ) ^ { 2 } + ( z - 4 ) ^ { 2 } = 4$
D) $( x + 2 ) ^ { 2 } + ( y - 3 ) ^ { 2 } + ( z - 4 ) ^ { 2 } = 16$

Question 10

Assume the segment with the endpoints $( 1 , - 2,1 )$ and $( 4,2,2 )$ is a diameter of a sphere. Give an equation of the sphere.

Question 11

Assume the segment with the endpoints $( 1,2,3 )$ and $( 5 , - 4,5 )$ is a diameter of a sphere. Give an equation of the sphere.

Question 12

Assume the segment with the endpoints $( 1,2,1 )$ and $( - 1,6,3 )$ is a diameter of a sphere. Give an equation of the sphere.

A) $( x - 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } + ( z - 2 ) ^ { 2 } = 24$
B) $x ^ { 2 } + ( y - 4 ) ^ { 2 } + ( z - 2 ) ^ { 2 } = 6$
C) $x ^ { 2 } + ( y - 4 ) ^ { 2 } + ( z - 2 ) ^ { 2 } = 2 \sqrt { 6 }$
D) $x ^ { 2 } + ( y - 4 ) ^ { 2 } + ( z - 2 ) ^ { 2 } = 12$

Question 13

If $( 1,2,3 )$ is the midpoint of the segment with one endpoint $( 1,1 , - 1 )$ , find the second endpoint.

Question 14

If $( 3,3 , - 3 )$ is the midpoint of the segment with one endpoint $( 1,2 , - 2 )$ , find the second endpoint.

Question 15

If $( 2,3,1 )$ is the midpoint of the segment with one endpoint $( 1,3,1 )$ , find the second endpoint.

A) $( 3,3,1 )$
B) $\left( \frac { 3 } { 2 } , 3,1 \right)$
C) $( 1,0,0 )$
D) $( 0 , - 1,3 )$

Question 16

Find the center and radius of the sphere with the equation $x ^ { 2 } - 2 x + y ^ { 2 } - 4 y + z ^ { 2 } + 1 = 0$

Question 17

Find the center and radius of the sphere with the equation $x ^ { 2 } - 2 x + y ^ { 2 } - 2 y + z ^ { 2 } + 2 z - 6 = 0$

Question 18

Find the center and radius of the sphere with the equation $x ^ { 2 } + 2 x + y ^ { 2 } - 2 y + z ^ { 2 } - 4 z + 5 = 0$

A) $( - 1,1,2 ) , r = 1$
B) $( 1 , - 1 , - 2 ) , r = 1$
C) $( - 1,1,2 ) , r = 5$
D) $( - 1,1,2 ) , r = \sqrt { 5 }$

Question 19

For $\vec { v } = \langle 1,2 \rangle \text { and } \vec { u } = \langle - 1,6 \rangle$ find $\vec { u } + \vec { v }$ and $\vec { u } - \vec { v }$

Question 20

For $\vec { v } = \langle - 1 , - 2 \rangle \text { and } \vec { u } = \langle 1,4 \rangle$ find $\vec { u } + \vec { v }$ and $\vec { u } - \vec { v }$

Question 21

For $\vec { v } = \langle 2,0 \rangle \text { and } \vec { u } = \langle 4,1 \rangle$ find $\vec { u } + \vec { v }$

A) $\langle 2 , - 1 \rangle$
B) $\langle 6 , - 1 \rangle$
C) $\langle 6,1 \rangle$
D) $\langle 2,1 \rangle$

Question 22

For $\vec { u } = \langle 1,2 , - 1 \rangle \text { and } \vec { v } = \langle 1,1 , - 2 \rangle$ find $\vec { u } + \vec { v }$ and $\vec { u } - \vec { v }$

Question 23

For $\vec { u } = \langle - 1 , - 1,4 \rangle \text { and } \vec { v } = \langle 1 , - 1,4 \rangle$ find $\vec { u } + \vec { v }$ and $\vec { u } - \vec { v }$

Question 24

For $\vec { u } = \langle 1 , - 2 , - 3 \rangle \text { and } \vec { v } = \langle - 1,1 , - 2 \rangle$ find $\vec { u } + \vec { v }$

A) $\langle 0 , - 1 , - 5 \rangle$
B) $\langle 2 , - 3 , - 1 \rangle$
C) $\langle 0 , - 3 , - 5 \rangle$
D) $\langle 0 , - 3 , - 3 \rangle$

Question 25

For $P = ( 2,1 ) \text { and } Q = ( 3,2 )$ , find $\overrightarrow { P Q }$

Question 26

For $P = ( - 1,3 ) \text { and } Q = ( 1 , - 1 )$ , find $\overrightarrow { P Q }$

Question 27

For $P = ( 1,2 ) \text { and } Q = ( 5,7 )$ , find $\overrightarrow { P Q }$

A) $\langle 6,9 \rangle$
B) $\langle - 4 , - 5 \rangle$
C) $\langle 8,7 \rangle$
D) $\langle 4,5 \rangle$

Question 28

For $P = ( 1,2,1 ) \text { and } Q = ( - 4,2,2 )$ , find $\overrightarrow { P Q }$

Question 29

For $P = ( - 1,3,9 ) \text { and } Q = ( 7,6,2 )$ , find $\overrightarrow { P Q }$

Question 30

For $P = ( 1,2,3 ) \text { and } Q = ( 3,3,5 )$ , find $\overrightarrow { P Q }$

A) $\langle 2,1,2 \rangle$
B) $\langle - 2 , - 1 , - 2 \rangle$
C) $\langle 4,5,8 \rangle$
D) $\langle - 4 , - 5 , - 8 \rangle$

Question 31

Find the norm of $\vec { u } = \langle 1,2 \rangle$

Question 32

Find the norm of $\vec { u } = \langle 2,1,3 \rangle$

Question 33

Find the norm of $\vec { u } = \langle 4,3,5 \rangle$

A) $25$
B) $2 \sqrt { 5 }$
C) $50$
D) $5 \sqrt { 2 }$

Question 34

Find a unit vector in the direction of $\vec { u } = \langle 4,3 \rangle$

Question 35

Find a unit vector in the direction of $\vec { u } = \langle 4,3 \rangle$

Question 36

Find a unit vector in the direction of $\vec { u } = \langle 1,2,3 \rangle$

Question 37

Find a unit vector in the direction of $\vec { u } = \langle 1,2,3 \rangle$

A) $\left\langle \frac { 1 } { 51 } , \frac { 2 } { 51 } , \frac { 3 } { 51 } \right\rangle$
B) $\left\langle \frac { 1 } { \sqrt { 14 } } , \frac { 2 } { \sqrt { 14 } } , \frac { 3 } { \sqrt { 14 } } \right\rangle$
C) $\langle 1,1 , \sqrt { 7 } \rangle$
D) $\left\langle \frac { 1 } { 9 } , \frac { 2 } { 9 } , \frac { 3 } { 9 } \right\rangle$

Question 38

Find a vector of length 10 parallel to $\langle 3,4 \rangle$

Question 39

Find the dot product of $\vec { u } = \langle 1,2 \rangle$ and $\vec { v } = \langle 2 , - 3 \rangle$ and the cosine of the angle between them.

Question 40

Find the dot product of $\vec { u } = \langle 2,3 \rangle$ and $\vec { v } = \langle - 3,2 \rangle$ and the cosine of the angle between them.

Question 41

Find the dot product of $\vec { u } = \langle 4,7 \rangle$ and $\vec { v } = \langle 1,3 \rangle$ and the cosine of the angle between them.

A) $10 , \cos ( \theta ) = \frac { 1 } { 13 }$
B) $25 , \cos ( \theta ) = \frac { 5 } { \sqrt { 26 } }$
C) $25 , \cos ( \theta ) = \frac { 1 } { 25 }$
D) $5 , \cos ( \theta ) = \frac { 5 } { \sqrt { 26 } }$

Question 42

Find the dot product of $\vec { u } = \langle 1,2,3 \rangle$ and $\vec { v } = \langle 2,1,1 \rangle$ and the cosine of the angle between them.

Question 43

Find the dot product of $\vec { u } = \langle - 1,2,4 \rangle$ and $\vec { v } = \langle 2 , - 4 , - 8 \rangle$ and the cosine of the angle between them.

Question 44

Find the dot product of $\vec { u } = \langle 1,2,1 \rangle$ and $\vec { v } = \langle 2,2,1 \rangle$ and the cosine of the angle between them.

A) $7 , \cos ( \theta ) = \frac { 7 } { 3 \sqrt { 6 } }$
B) $7 , \cos ( \theta ) = \frac { 7 } { 54 }$
C) $9 , \cos ( \theta ) = \frac { 1 } { 6 }$
D) $6 , \cos ( \theta ) = \frac { 1 } { 9 }$

Question 45

$\text { Find } \operatorname { comp } _ { \vec { u } } \vec { v } , \operatorname { proj } _ { \vec { u } } \vec { v } \text {, and the component of } \vec { v } \text { orthogonal to } \vec { u } \text {, where } \vec { u } = \langle 1,2 \rangle \text { and } \vec { v } = \langle - 1,3 \rangle \text {. }$

Question 46

Find $\operatorname { com } p _ { \vec { u } } \vec { v }$ , $\operatorname { proj } _ { \vec { u } } \vec { v }$ , and the component of $\vec { v }$ orthogonal to $\vec { u }$ , where $\vec { u } = \langle 1,2 \rangle \text { and } \vec { v } = \langle 2 , - 5 \rangle$

Question 47

Find $\operatorname { com } p _ { \vec { u } } \vec { v }$ , $\operatorname { proj } _ { \vec { u } } \vec { v }$ , and the component of $\vec { v }$ orthogonal to $\vec { u }$ , where $\vec { u } = \langle 1,1 \rangle \text { and } \vec { v } = \langle - 2,1 \rangle$

A) $c o m p _ { \vec { u } } \vec { v } = - \frac { 1 } { \sqrt { 2 } }$ $\operatorname { proj } _ { \vec { u } } \vec { v } = \left\langle - \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right\rangle$ orthogonal component is $\left\langle - \frac { 3 } { 2 } , \frac { 3 } { 2 } \right\rangle$
B) $c o m p _ { \vec { u } } \vec { v } = - \frac { 1 } { \sqrt { 2 } }$ $\operatorname { proj } _ { \vec { u } } \vec { v } = \left\langle - \frac { 1 } { 4 } , - \frac { 1 } { 4 } \right\rangle$ orthogonal component is $\left\langle - \frac { 3 } { 2 } , \frac { 3 } { 4 } \right\rangle$
C) $\operatorname { comp }_{ \overrightarrow { \vec { u }} } \vec { v } = - \frac { 1 } { \sqrt { 3 } }$ $\operatorname { proj } _ { \vec { u } } \vec { v } = \left\langle - \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right\rangle$ orthogonal component is $\left\langle - \frac { 3 } { 4 } , \frac { 3 } { 2 } \right\rangle$
D) $\operatorname { comp }_{ \overrightarrow { \vec { u } }} \vec { v } = - \frac { 1 } { \sqrt { 3 } }$ $\operatorname { proj } _ { \vec { u } } \vec { v } = \left\langle - \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right\rangle$ orthogonal component is $\left\langle - \frac { 3 } { 4 } , \frac { 3 } { 2 } \right\rangle$

Question 48

Find $\operatorname { com } p _ { \vec { u } } \vec { v }$ , $\operatorname { proj } _ { \vec { u } } \vec { v }$ , and the component of $\vec { v }$ orthogonal to $\vec { u }$ , where $\vec { u } = \langle 1,2,1 \rangle \text { and } \vec { v } = \langle - 1,3,2 \rangle$

Question 49

Find $\operatorname { com } p _ { \vec { u } } \vec { v }$ , $\operatorname { proj } _ { \vec { u } } \vec { v }$ , and the component of $\vec { v }$ orthogonal to $\vec { u }$ , where $\vec { u } = \langle 2,2,1 \rangle \text { and } \vec { v } = \langle 1 , - 3,4 \rangle$

Question 50

Find $\operatorname { com } p _ { \vec { u } } \vec { v }$ , $\operatorname { proj } _ { \vec { u } } \vec { v }$ , and the component of $\vec { v }$ orthogonal to $\vec { u }$ , where $\vec { u } = \langle 1,1,1 \rangle \text { and } \vec { v } = \langle 3,2,1 \rangle$

Question 51

Find $\operatorname { com } p _ { \vec { u } } \vec { v }$ , $\operatorname { proj } _ { \vec { u } } \vec { v }$ , and the component of $\vec { v }$ orthogonal to $\vec { u }$ , where $\vec { u } = \langle 1,2,1 \rangle \text { and } \vec { v } = \langle - 1,1 , - 1 \rangle$

A) $\operatorname { comp } p _ { \vec { u } } \vec { v } = 0$ $\operatorname { proj } _ { \vec { u } } \vec { v } = \langle 2,1,1 \rangle$ orthogonal component is $\langle 0,0,0 \rangle$
B) $\operatorname { comp } p _ { \vec { u } } \vec { v } = 0$ $\operatorname { proj } _ { \vec { u } } \vec { v } = \langle 0,0,0 \rangle$ orthogonal component is $\langle - 1,1 , - 1 \rangle$
C) $\operatorname { comp } _ { \vec { u } } \vec { v } = 1$ $\operatorname { proj } _ { \vec { u } } \vec { v } = \langle 0,0,0 \rangle$ orthogonal component is $\langle 2,1,1 \rangle$
D) $\operatorname { comp } p _ { \vec { u } } \vec { v } = 0$ $\operatorname { proj } _ { \vec { u } } \vec { v } = \langle 0,0,0 \rangle$ orthogonal component is $\langle 1,2,1 \rangle$

Question 52

For $\vec { u } = \langle 1,2,3 \rangle$ and $\vec { v } = \langle - 1,1 , - 1 \rangle$ $\vec { w } = \langle 1,2,1 \rangle$ , find $\vec { u } \times \vec { v }$

Question 53

For $\vec { u } = \langle 1,2,3 \rangle$ , $\vec { v } = \langle - 1,1 , - 1 \rangle$ , and $\vec { w } = \langle 1,2,1 \rangle$ , find $\vec { v } \times \vec { u }$

Question 54

For $\vec { u } = \langle 1,2,3 \rangle$ , $\vec { v } = \langle - 1,1 , - 1 \rangle$ , and $\vec { w } = \langle 1,2,1 \rangle$ , find $( \vec { u } \times \vec { v } ) \times \vec { w }$

Question 55

For $\vec { u } = \langle 1,2,3 \rangle$ , $\vec { v } = \langle - 1,1 , - 1 \rangle$ and $\vec { w } = \langle 1,2,1 \rangle$ , find $( \vec { u } \times \vec { v } ) \times \vec { u }$

A) $\langle - 4,2,0 \rangle$
B) $\langle 0,0,0 \rangle$
C) $\langle - 4,2,1 \rangle$
D) $\langle - 4,2 , - 1 \rangle$

Question 56

For $\vec { u } = \langle 1,2,3 \rangle$ , $\vec { v } = \langle - 1,1 , - 1 \rangle$ , and $\vec { w } = \langle 1,2,1 \rangle$ , find $\vec { u } \cdot ( \vec { v } \times \vec { w } )$

Question 57

For $\vec { u } = \langle 1,2,3 \rangle$ , $\vec { v } = \langle - 1,1 , - 1 \rangle$ , and $\vec { w } = \langle 1,2,1 \rangle$ , find $\vec { v } \cdot ( \vec { u } \times \vec { w } )$

A) -6
B) 6
C) $\langle - 4,2,0 \rangle$
D) $\langle 4 , - 2,0 \rangle$

Question 58

Find the area of the parallelogram determined by $\vec { u } = \langle 1,2,3 \rangle$ and $\vec { w } = \langle 1,2,1 \rangle$

Question 59

Find the area of the parallelogram determined by $\vec { u } = \langle 1,2,3 \rangle$ and $\vec { w } = \langle 1,1,1 \rangle$

Question 60

Find the area of the parallelogram determined by $\vec { v } = \langle - 1,1 , - 1 \rangle$ and $\vec { w } = \langle 1,2,1 \rangle$

A) $2 \sqrt { 3 }$
B) 12
C) $3 \sqrt { 2 }$
D) $6$

Question 61

Find the volume of the parallelepiped determined by $\vec { u } = \langle 1,2,3 \rangle$ , $\vec { v } = \langle - 1,1 , - 1 \rangle$ , and $\vec { w } = \langle 1,2,1 \rangle$

Question 62

Find the area of the triangle with vertices $P ( 0,0,0 ) , Q ( 1,2,3 ) , \text { and } R ( 1,2,1 )$

Question 63

Find a vector perpendicular to the plane determined by the points $P ( 1,2,3 ) , Q ( 4,5,6 ) \text {, and } R ( 1,1 , - 1 )$ , whose third coordinate is 1.

Question 64

Find a unit vector perpendicular to the plane determined by the points $P ( 1,2,3 ) , Q ( 4,5,6 ) \text {, and } R ( 1,1 , - 1 )$

A) $\left\langle \frac { 3 } { \sqrt { 26 } } , - \frac { 4 } { \sqrt { 26 } } , \frac { 1 } { \sqrt { 26 } } \right\rangle$
B) $< \frac { 3 } { \sqrt { 26 } } , \frac { 4 } { \sqrt { 26 } } , \frac { 1 } { \sqrt { 26 } } >$
C) $< \frac { 3 } { 26 } , - \frac { 4 } { 26 } , \frac { 1 } { 26 } >$
D) $\left. < \frac { 3 } { 26 } , \frac { 4 } { 26 } , \frac { 1 } { 26 } \right\rangle$

Question 65

Find the area of the triangle in the plane with vertices $P ( 1,2 ) , Q ( 4,6 ) , \text { and } R ( - 1 , - 1 )$

Question 66

Give an equation of the line containing the points $( 1,3,2 ) \text { and } ( 2,1,4 )$ as vector parameterization.

Question 67

Give the parametric equations of the line containing the points $( 1,3,2 ) \text { and } ( 2,1,4 )$

Question 68

Give the symmetric equations of the line containing the points $( 1,3,2 ) \text { and } ( 2,1,4 )$

Question 69

Give an equation of the line containing the points $( 2,1,1 ) \text { and } ( 4,3,2 )$ as vector parameterization.
A) $\vec { r } ( t ) = \langle 1 , - 2,2 \rangle + t \langle 1,3,2 \rangle$
B) $\vec { r } ( t ) = \langle 2,1,1 \rangle + t \langle 2,2,1 \rangle$
C) $\vec { r } ( t ) = \langle 1,3,2 \rangle + t \langle 1,2,2 \rangle$
D) $\vec { r } ( t ) = \langle 1,2,2 \rangle + t \langle 1,3,2 \rangle$

Question 70

Give a set of parametric equations for the line containing the points $( 2,1,1 ) \text { and } ( 4,3,2 )$

A) $\begin{array} { l } x = 1 + t \\y = - 2 + 3 t \\z = 2 + 2 t\end{array}$
B) $\begin{array} { l } x = 2 + 2 t \\y = 1 + 2 t \\z = 1 + t\end{array}$
C) $\begin{array} { l } x = 1 + t \\y = 3 + 2 t \\z = 2 + 2 t\end{array}$
D) $\begin{array} { l } x = 1 + t \\y = 2 + 3 t \\z = 2 + 2 t\end{array}$

Question 71

Give a set of symmetric equations for the line containing the points $( 2,1,1 ) \text { and } ( 4,3,2 )$

A) $\frac { x - 2 } { 2 } = \frac { y - 1 } { 2 } = z - 1$
B) $x - 1 = \frac { 3 - y } { 2 } = \frac { z - 2 } { 2 }$
C) $x - 1 = \frac { y - 3 } { 2 } = \frac { z - 2 } { 2 }$
D) $x - 1 = \frac { y - 2 } { 3 } = \frac { z - 2 } { 2 }$

Question 72

Give an equation of the line containing the points $( 2,2 - 3 ) \text { and } ( 6,1,2 )$ as vector parameterization.

Question 73

Give the parametric equations of the line containing the points $( 2,2 - 3 ) \text { and } ( 6,1,2 )$

Question 74

Give the symmetric equations of the line containing the points $( 2,2 - 3 ) \text { and } ( 6,1,2 )$

Question 75

Give an equation of the line containing the point $( 1,2 , - 1 )$ and parallel to $\vec { d } - \langle 2 , - 3,1 \rangle$ as vector parameterization.

Question 76

Give the parametric equations of the line containing the point $( 1,2 , - 1 )$ and parallel to $\vec { d } = \langle 2 , - 3,1 \rangle$

Question 77

Give the symmetric equations of the line containing the point $( 1,2 , - 1 )$ and parallel to $\vec { d } = \langle 2 , - 3,1 \rangle$

Question 78

Find the distance from the point $( 1,1,2 )$ to the line $\vec { r } ( t ) = \langle 0 , - 1,3 \rangle + t \langle 1,2,1 \rangle$

Question 79

Are the given lines parallel, intersecting, identical or skew? $\vec { r } _ { 1 } ( t ) = \langle 2,3,6 \rangle + t \langle - 1,2,4 \rangle$ $\vec { r } _ { 2 } ( t ) = \langle 9 , - 1 , - 2 \rangle + t \langle 2 , - 4 , - 8 \rangle$

A) Parallel
B) Identical
C) Intersecting
D) Skew

Question 80

Find an equation for the plane containing the point $( 1,1,2 )$ and normal to the vector $\langle 1 , - 1,3 \rangle$