Question 1

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

Accepted Answer

$\vec { r } ( t ) = \langle 2,2 , - 3 \rangle + t \langle 4 , - 1,5 \rangle$ Note that there are other forms.

Question 2

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

Accepted Answer

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

Question 3

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

Accepted Answer

$( x - 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } + ( z - 1 ) ^ { 2 } = 4$

Question 4

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.

Accepted Answer

The answer of Find the dot product of \(\vec {...

Question 5

Find an equation for the plane that contains the point $( 1,2 , - 1 )$ and the line $\vec { r } ( t ) = \langle 1,3,4 \rangle + t \langle 1,1,3 \rangle$

Accepted Answer

The answer of Find an equation for the plane that...

Question 6

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

Accepted Answer

The answer of Find a unit vector in the direction...

Question 7

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

Accepted Answer

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

Question 8

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

Accepted Answer

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 $\left\langle - \frac { 3 } { 4 } , \frac { 3 } { 2 } \right\rangle$ 
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 9

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) $\begin{array} { l } 
x = 1 + t \
y = 2 + 3 t \
z = 2 + 2 t
\end{array}$

Accepted Answer

The answer of Give a set of parametric equations for...

Question 10

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

Accepted Answer

The answer of Find a unit vector in the direction...

Question 11

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

Accepted Answer

The answer of Find the area of the parallelogram determined...

Question 12

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

Accepted Answer

The answer of For \(\vec { u } = \langle...

Question 13

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

Accepted Answer

A)  $\langle 0 , - 1 , - 5 \rangle$ 
B)  $\langle 2 , - 3 , - 1 \rangle$ 
C)  $\langle 0 , - 3 , - 5 \rangle$ 
D)  $\langle 0 , - 3 , - 3 \rangle$ 
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 14

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

Accepted Answer

The answer of Find a unit vector in the direction...

Question 15

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

Accepted Answer

@#LAT-DLM& Note that

Question 16

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

Accepted Answer

The answer of What is the distance between the given...

Question 17

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

Accepted Answer

Note that

Question 18

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

Accepted Answer

The answer of For \(\vec { u } = \langle...

Question 19

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

Accepted Answer

The answer of If $( 1,2,3 )$ is the midpoint...

Question 20

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.

Accepted Answer

The answer of Find the dot product of \(\vec {...

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

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

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

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.

Find an equation for the plane that contains the point $( 1,2 , - 1 )$ and the line $\vec { r } ( t ) = \langle 1,3,4 \rangle + t \langle 1,1,3 \rangle$

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

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

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

Give a set of parametric equations for the line containing the points $( 2,1,1 ) \text { and } ( 4,3,2 )$ A) x=1+t y=-2+3t z=2+2t B) x=2+2t y=1+2t z=1+t C) x=1+t y=3+2t z=2+2t D) x=1+t y=2+3t z=2+2t

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

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

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

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

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

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

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

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

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

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

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.

Limits

Derivatives

Applications of the Derivative

Definite Integrals

Techniques of Integration

Applications of Integration

Sequences and Series

Power Series

Parametric Equations Polar Coordinates and Conic Sections

Vector Functions

Multivariable Functions

Double and Triple Integrals

Vector Analysis

Functions and Precalculus

Filters

Exam 10: Vectors

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

For v⃗=⟨2,0⟩ and u⃗=⟨4,1⟩\vec { v } = \langle 2,0 \rangle \text { and } \vec { u } = \langle 4,1 \ranglev=⟨2,0⟩ and u=⟨4,1⟩ find u⃗+v⃗\vec { u } + \vec { v }u+v

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

Find the dot product of u⃗=⟨1,2⟩\vec { u } = \langle 1,2 \rangleu=⟨1,2⟩ and v⃗=⟨2,−3⟩\vec { v } = \langle 2 , - 3 \ranglev=⟨2,−3⟩ and the cosine of the angle between them.

Find an equation for the plane that contains the point (1,2,−1)( 1,2 , - 1 )(1,2,−1) and the line r⃗(t)=⟨1,3,4⟩+t⟨1,1,3⟩\vec { r } ( t ) = \langle 1,3,4 \rangle + t \langle 1,1,3 \rangler(t)=⟨1,3,4⟩+t⟨1,1,3⟩

Find a unit vector in the direction of u⃗=⟨4,3⟩\vec { u } = \langle 4,3 \rangleu=⟨4,3⟩

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

Give a set of parametric equations for the line containing the points (2,1,1) and (4,3,2)( 2,1,1 ) \text { and } ( 4,3,2 )(2,1,1) and (4,3,2) A) x=1+t y=-2+3t z=2+2t B) x=2+2t y=1+2t z=1+t C) x=1+t y=3+2t z=2+2t D) x=1+t y=2+3t z=2+2t

Find a unit vector in the direction of u⃗=⟨1,2,3⟩\vec { u } = \langle 1,2,3 \rangleu=⟨1,2,3⟩

Find the area of the parallelogram determined by u⃗=⟨1,2,3⟩\vec { u } = \langle 1,2,3 \rangleu=⟨1,2,3⟩ and w⃗=⟨1,1,1⟩\vec { w } = \langle 1,1,1 \ranglew=⟨1,1,1⟩

For u⃗=⟨1,−2,−3⟩ and v⃗=⟨−1,1,−2⟩\vec { u } = \langle 1 , - 2 , - 3 \rangle \text { and } \vec { v } = \langle - 1,1 , - 2 \rangleu=⟨1,−2,−3⟩ and v=⟨−1,1,−2⟩ find u⃗+v⃗\vec { u } + \vec { v }u+v

Find a unit vector in the direction of u⃗=⟨4,3⟩\vec { u } = \langle 4,3 \rangleu=⟨4,3⟩

Give the symmetric equations of the line containing the point (1,2,−1)( 1,2 , - 1 )(1,2,−1) and parallel to d⃗=⟨2,−3,1⟩\vec { d } = \langle 2 , - 3,1 \rangled=⟨2,−3,1⟩

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

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

For u⃗=⟨1,2,3⟩\vec { u } = \langle 1,2,3 \rangleu=⟨1,2,3⟩ , v⃗=⟨−1,1,−1⟩\vec { v } = \langle - 1,1 , - 1 \ranglev=⟨−1,1,−1⟩ , and w⃗=⟨1,2,1⟩\vec { w } = \langle 1,2,1 \ranglew=⟨1,2,1⟩ , find v⃗×u⃗\vec { v } \times \vec { u }v×u

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

Find the dot product of u⃗=⟨1,2,3⟩\vec { u } = \langle 1,2,3 \rangleu=⟨1,2,3⟩ and v⃗=⟨2,1,1⟩\vec { v } = \langle 2,1,1 \ranglev=⟨2,1,1⟩ and the cosine of the angle between them.

Limits

Derivatives

Applications of the Derivative

Definite Integrals

Techniques of Integration

Applications of Integration

Sequences and Series

Power Series

Parametric Equations Polar Coordinates and Conic Sections

Vector Functions

Multivariable Functions

Double and Triple Integrals

Vector Analysis

Functions and Precalculus

Filters

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

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

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

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.

Find an equation for the plane that contains the point $( 1,2 , - 1 )$ and the line $\vec { r } ( t ) = \langle 1,3,4 \rangle + t \langle 1,1,3 \rangle$

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

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

Give a set of parametric equations for the line containing the points $( 2,1,1 ) \text { and } ( 4,3,2 )$ A) x=1+t y=-2+3t z=2+2t B) x=2+2t y=1+2t z=1+t C) x=1+t y=3+2t z=2+2t D) x=1+t y=2+3t z=2+2t

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

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

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

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

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

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

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

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

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

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.