Question 1

Find the direction of the vector $v = 6 i + 6 \sqrt { 3 } \mathbf { j }$

Accepted Answer

A)  $\theta = 30 ^ { \circ }$ 
B)  $\theta = 150 ^ { \circ }$ 
C)  $\theta = 120 ^ { \circ }$ 
D)  $\theta = 210 ^ { \circ }$ 
E)  $\theta = 60 ^ { \circ }$ 
A)  $\theta = 30 ^ { \circ }$ 
B)  $\theta = 150 ^ { \circ }$ 
C)  $\theta = 120 ^ { \circ }$ 
D)  $\theta = 210 ^ { \circ }$ 
E)  $\theta = 60 ^ { \circ }$ E

Question 2

Find the vector $\mathbf { Y }$ having length $| \mathrm { v } | = 2$
, and direction $\theta = 135 ^ { \circ }$

Accepted Answer

A)  $\mathrm { v } = \langle - 1,1 \rangle$ 
B)  $\boldsymbol { v } = \langle 1 , \sqrt { 2 } \rangle$ 
C)  $\mathrm { v } = \langle - \sqrt { 2 } , \sqrt { 2 } \rangle$ 
D)  $\mathbf { v } = \langle 1,1 \rangle$ 
E)  $\mathrm { v } = \langle - 2,2 \rangle$ 
A)  $\mathrm { v } = \langle - 1,1 \rangle$ 
B)  $\boldsymbol { v } = \langle 1 , \sqrt { 2 } \rangle$ 
C)  $\mathrm { v } = \langle - \sqrt { 2 } , \sqrt { 2 } \rangle$ 
D)  $\mathbf { v } = \langle 1,1 \rangle$ 
E)  $\mathrm { v } = \langle - 2,2 \rangle$ C

Question 3

Given $\mathbf { u } = \langle 2,1 \rangle$ 
and $\mathbf { v } = \langle - 3,1 \rangle$ 
, calculate $\operatorname { proj } _ { \mathbf { x } } \mathbf { u }$ 
and then resolve $\mathbf { u } $ 
into $\mathbf { u } _ { 1 }$ 
and $\mathbf { u } _ { 2 }$ 
.

Accepted Answer

$\operatorname { proj } _ { \mathbf { v } } \mathbf { u } = \left( \frac { \mathbf { u } \cdot \mathbf { v } } { | \mathbf { v } | ^ { 2 } } \right) \mathbf { v } = \left( \frac { ( 2 ) ( - 3 ) + ( 1 ) ( 1 ) } { ( - 3 ) ^ { 2 } + ( 1 ) ^ { 2 } } \right) ( - 3,1 ) = - \frac { 1 } { 2 } ( - 3,1 ) = \left( \frac { 3 } { 2 } , - \frac { 1 } { 2 } \right)$ 
. Thus $\mathbf { u } _ { 1 } = \left( \frac { 3 } { 2 } , - \frac { 1 } { 2 } \right)$ 
, $\mathbf { u } _ { 2 } = ( 2,1 ) - \left( \frac { J } { 2 } , - \frac { 1 } { 2 } \right) = \left( \frac { 1 } { 2 } , \frac { 3 } { 2 } \right)$ 
.

Question 4

Given $\mathbf { u } = ( 4 , - 3 )$ and $\mathrm { v } = \langle , 2,2 \rangle$
, calculate $\mathrm { proj } _ { \mathbf { v } } \mathbf { u }$

Accepted Answer

A)  $\frac { 14 } { 17 } , - \frac { 6 ] } { 17 }$ 
B)  $\frac { 34 } { 17 } , \frac { 12 } { 17 }$ 
C)  $\langle 54,12 \rangle$ 
D)  $\left( 17 , \frac { 12 } { 17 } \right)$ 
E)  none 
A)  $\frac { 14 } { 17 } , - \frac { 6 ] } { 17 }$ 
B)  $\frac { 34 } { 17 } , \frac { 12 } { 17 }$ 
C)  $\langle 54,12 \rangle$ 
D)  $\left( 17 , \frac { 12 } { 17 } \right)$ 
E)  none

Question 5

A man pulls a sled horizontally by exerting $641 b$ on the rope that is tied to its front end. If the rope makes an angle of $45 ^ { \circ }$
With horizontal, find the work done in moving the sled $35$
Ft

Accepted Answer

A)  $2929.17$
Ft-lb 
B)  $629.17$
Ft-lb 
C)  $1583.92$
Ft-lb 
D)  $630.25$
Ft-lb 
E)  $1829.17$
Ft-lb 
A)  $2929.17$
Ft-lb 
B)  $629.17$
Ft-lb 
C)  $1583.92$
Ft-lb 
D)  $630.25$
Ft-lb 
E)  $1829.17$
Ft-lb

Question 6

If the vector v has initial point P, what is its terminal point? $\mathbf { v } = ( 0,0,1 ) , P ( 1,1 , - 1 )$

Accepted Answer

The answer of If the vector v has initial point...

Question 7

Two vectors u and v are given. Find the angle to one decimal place (expressed in degrees) between u and v. $\mathbf { u } = \langle 4,0 , - 2 \rangle , \mathbf { v } = \langle 2,1,0 \rangle$

Accepted Answer

The answer of Two vectors u and v are given....

Question 8

Find parametric equations for the line that passes through the point P and is parallel to the vector v. $P ( 0,0,0 ) , \quad \mathrm { v } = \langle - 5,2,9 \rangle$

Accepted Answer

A)  $x = - 5 t , y = 2 t , z = 9 t$ 
B)  $x = 5 - 5 t , y = 2 + 2 t , z = 9 - 9 t$ 
C)  $x = 2 - 5 t , y = - 5 + 2 t , z = 9 t$ 
D)  $x = - 5 , y = 2 , z = 9$ 
E)  none of these 
A)  $x = - 5 t , y = 2 t , z = 9 t$ 
B)  $x = 5 - 5 t , y = 2 + 2 t , z = 9 - 9 t$ 
C)  $x = 2 - 5 t , y = - 5 + 2 t , z = 9 t$ 
D)  $x = - 5 , y = 2 , z = 9$ 
E)  none of these

Question 9

Find parametric equations for the line that passes through the points P and Q. $P ( 2 , - 1 , - 1 ) , \quad Q ( 0,1 , - 3 )$

Accepted Answer

The answer of Find parametric equations for the line that...

Question 10

Find the center and radius of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x + 4 - 8 y = 0$

Accepted Answer

A)  $C ( 2,2,4 ) ; r = \sqrt { 2 }$ 
B)  $C ( 0,2,4 ) ; r = 4$ 
C)  $C ( 2,4,0 ) ; r= \sqrt { 2 }$ 
D)  $C ( 2,4,0 ) ;r= 2$ 
E) none of these 
A)  $C ( 2,2,4 ) ; r = \sqrt { 2 }$ 
B)  $C ( 0,2,4 ) ; r = 4$ 
C)  $C ( 2,4,0 ) ; r= \sqrt { 2 }$ 
D)  $C ( 2,4,0 ) ;r= 2$ 
E) none of these

Question 11

Find $\mathbf { u} \cdot \mathbf { Y }$ . $\mathbf { u } = \langle 5,0 ) , \mathbf { v } = ( \sqrt { 3 } , 1 )$

Accepted Answer

A)  $16$ 
B)  $- 16$ 
C)  $5 \sqrt { 3 } + 1$ 
D)  $5 \sqrt { 3 }$ 
E)  $5$ 
A)  $16$ 
B)  $- 16$ 
C)  $5 \sqrt { 3 } + 1$ 
D)  $5 \sqrt { 3 }$ 
E)  $5$

Question 12

A plane has normal vector n and passes through the point P. Find an equation for the plane. $\mathbf { n } = 5 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } , P ( 0,2 , - 3 )$

Accepted Answer

The answer of A plane has normal vector n and...

Question 13

Given that the forces $\mathbf { F } _ { 1 } = ( - 10,3 )$ , $\mathbf { F } _ { 2 } = ( - 4,1 )$
And $\mathbf { F } _ { 3 } = \langle 4 , - 10 \rangle$
Are acting on a point $p$
, find the resultant force

Accepted Answer

A)  $( - 6,14 )$ 
B)  $( - 10 , - 6 )$ 
C)  $( - 1 , - 6 )$ 
D)  $( - 18,6 )$ 
E)  $( 18 , - 6 )$ 
A)  $( - 6,14 )$ 
B)  $( - 10 , - 6 )$ 
C)  $( - 1 , - 6 )$ 
D)  $( - 18,6 )$ 
E)  $( 18 , - 6 )$

Question 14

A plane has normal vector n and passes through the point P. Find an equation for the plane. $\mathbf { n } = \left( 3,0 , - \frac { 1 } { 3 } \right) , P ( 2,4,9 )$

Accepted Answer

The answer of A plane has normal vector n and...

Question 15

Find the component of $\mathbf { u } $
 
along $\mathbf { Y }$ 
if $\mathbf { u } = - \frac { 1 } { 2 } \mathbf { i } + 2 \mathbf { j }$ 
and $\mathbf { v } = - 2 \mathbf { i } + \frac { 1 } { 2 } \mathbf { j }$ 
.

Accepted Answer

The answer of Find the component of \(\mathbf { u...

Question 16

A water tank is in the shape of a sphere of radius 10 feet. The tank is supported on a metal circle 8 feet below the center of the sphere, as shown in the figure. Find the radius of the metal circle.

Accepted Answer

The answer of A water tank is in the shape...

Question 17

Find the work done by the force $\mathbf { F } = 5 \mathbf { i } + 25 \mathbf { j }$ 
in moving an object from $P ( 0,0 )$ 
to $Q ( 1,6 )$ 
.

Accepted Answer

The answer of Find the work done by the force...

Question 18

Determine whether $\mathbf { u } = 2 \sqrt { 3 } \mathbf { i } + 2 \mathbf { j }$ 
is orthogonal to $\mathbf { v } = \mathbf { i } - \sqrt { 3 } \mathbf { j }$ 
. Justify your answer.

Accepted Answer

The answer of Determine whether \(\mathbf { u } =...

Question 19

Find the area of the parallelogram determined by the given vectors. $\mathbf { u } = ( 0 , - 2,3 ) , \mathbf { v } = \langle 5 , - 5,0 )$

Accepted Answer

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

Question 20

Find the work done by the force $\mathbf { F } = 100 \mathbf { i } - 25 \mathbf { j }$ 
in moving an object from $P ( - 2,4 )$ 
to $Q ( 150,5 )$ 
.

Accepted Answer

The answer of Find the work done by the force...

Find the direction of the vector $v = 6 i + 6 \sqrt { 3 } \mathbf { j }$

Find the vector $\mathbf { Y }$ having length $| \mathrm { v } | = 2$ , and direction $\theta = 135 ^ { \circ }$

Given $\mathbf { u } = \langle 2,1 \rangle$ and $\mathbf { v } = \langle - 3,1 \rangle$ , calculate $\operatorname { proj } _ { \mathbf { x } } \mathbf { u }$ and then resolve $\mathbf { u }$ into $\mathbf { u } _ { 1 }$ and $\mathbf { u } _ { 2 }$ .

Given $\mathbf { u } = ( 4 , - 3 )$ and $\mathrm { v } = \langle , 2,2 \rangle$ , calculate $\mathrm { proj } _ { \mathbf { v } } \mathbf { u }$

A man pulls a sled horizontally by exerting $641 b$ on the rope that is tied to its front end. If the rope makes an angle of $45 ^ { \circ }$ With horizontal, find the work done in moving the sled $35$ Ft

If the vector v has initial point P, what is its terminal point? $\mathbf { v } = ( 0,0,1 ) , P ( 1,1 , - 1 )$

Two vectors u and v are given. Find the angle to one decimal place (expressed in degrees) between u and v. $\mathbf { u } = \langle 4,0 , - 2 \rangle , \mathbf { v } = \langle 2,1,0 \rangle$

Find parametric equations for the line that passes through the point P and is parallel to the vector v. $P ( 0,0,0 ) , \quad \mathrm { v } = \langle - 5,2,9 \rangle$

Find parametric equations for the line that passes through the points P and Q. $P ( 2 , - 1 , - 1 ) , \quad Q ( 0,1 , - 3 )$

Find the center and radius of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x + 4 - 8 y = 0$

Find $\mathbf { u} \cdot \mathbf { Y }$ . $\mathbf { u } = \langle 5,0 ) , \mathbf { v } = ( \sqrt { 3 } , 1 )$

A plane has normal vector n and passes through the point P. Find an equation for the plane. $\mathbf { n } = 5 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } , P ( 0,2 , - 3 )$

Given that the forces $\mathbf { F } _ { 1 } = ( - 10,3 )$ , $\mathbf { F } _ { 2 } = ( - 4,1 )$ And $\mathbf { F } _ { 3 } = \langle 4 , - 10 \rangle$ Are acting on a point $p$ , find the resultant force

A plane has normal vector n and passes through the point P. Find an equation for the plane. $\mathbf { n } = \left( 3,0 , - \frac { 1 } { 3 } \right) , P ( 2,4,9 )$

Find the component of $\mathbf { u }$ along $\mathbf { Y }$ if $\mathbf { u } = - \frac { 1 } { 2 } \mathbf { i } + 2 \mathbf { j }$ and $\mathbf { v } = - 2 \mathbf { i } + \frac { 1 } { 2 } \mathbf { j }$ .

A water tank is in the shape of a sphere of radius 10 feet. The tank is supported on a metal circle 8 feet below the center of the sphere, as shown in the figure. Find the radius of the metal circle.

Find the work done by the force $\mathbf { F } = 5 \mathbf { i } + 25 \mathbf { j }$ in moving an object from $P ( 0,0 )$ to $Q ( 1,6 )$ .

Determine whether $\mathbf { u } = 2 \sqrt { 3 } \mathbf { i } + 2 \mathbf { j }$ is orthogonal to $\mathbf { v } = \mathbf { i } - \sqrt { 3 } \mathbf { j }$ . Justify your answer.

Find the area of the parallelogram determined by the given vectors. $\mathbf { u } = ( 0 , - 2,3 ) , \mathbf { v } = \langle 5 , - 5,0 )$

Find the work done by the force $\mathbf { F } = 100 \mathbf { i } - 25 \mathbf { j }$ in moving an object from $P ( - 2,4 )$ to $Q ( 150,5 )$ .

Equations and Graphs

Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions: Right Triangle Approach

Trigonometric Functions: Unit Circle Approach

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Polar Coordinates and Parametric Equations

Vectors in Two and Three Dimensions

Matrices and Determinants

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Exam 10: Systems of Equations and Inequalities

Find the direction of the vector v=6i+63jv = 6 i + 6 \sqrt { 3 } \mathbf { j }v=6i+63​j

Find the vector Y\mathbf { Y }Y having length ∣v∣=2| \mathrm { v } | = 2∣v∣=2 , and direction θ=135∘\theta = 135 ^ { \circ }θ=135∘

Given u=(4,−3)\mathbf { u } = ( 4 , - 3 )u=(4,−3) and v=⟨,2,2⟩\mathrm { v } = \langle , 2,2 \ranglev=⟨,2,2⟩ , calculate projvu\mathrm { proj } _ { \mathbf { v } } \mathbf { u }projv​u

A man pulls a sled horizontally by exerting 641b641 b641b on the rope that is tied to its front end. If the rope makes an angle of 45∘45 ^ { \circ }45∘ With horizontal, find the work done in moving the sled 353535 Ft

If the vector v has initial point P, what is its terminal point? v=(0,0,1),P(1,1,−1)\mathbf { v } = ( 0,0,1 ) , P ( 1,1 , - 1 )v=(0,0,1),P(1,1,−1)

Two vectors u and v are given. Find the angle to one decimal place (expressed in degrees) between u and v. u=⟨4,0,−2⟩,v=⟨2,1,0⟩\mathbf { u } = \langle 4,0 , - 2 \rangle , \mathbf { v } = \langle 2,1,0 \rangleu=⟨4,0,−2⟩,v=⟨2,1,0⟩

Find parametric equations for the line that passes through the point P and is parallel to the vector v. P(0,0,0),v=⟨−5,2,9⟩P ( 0,0,0 ) , \quad \mathrm { v } = \langle - 5,2,9 \rangleP(0,0,0),v=⟨−5,2,9⟩

Find parametric equations for the line that passes through the points P and Q. P(2,−1,−1),Q(0,1,−3)P ( 2 , - 1 , - 1 ) , \quad Q ( 0,1 , - 3 )P(2,−1,−1),Q(0,1,−3)

Find the center and radius of the sphere. x2+y2+z2−4x+4−8y=0x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x + 4 - 8 y = 0x2+y2+z2−4x+4−8y=0

Find u⋅Y\mathbf { u} \cdot \mathbf { Y }u⋅Y . u=⟨5,0),v=(3,1)\mathbf { u } = \langle 5,0 ) , \mathbf { v } = ( \sqrt { 3 } , 1 )u=⟨5,0),v=(3​,1)

A plane has normal vector n and passes through the point P. Find an equation for the plane. n=5i−j+2k,P(0,2,−3)\mathbf { n } = 5 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } , P ( 0,2 , - 3 )n=5i−j+2k,P(0,2,−3)

Given that the forces F1=(−10,3)\mathbf { F } _ { 1 } = ( - 10,3 )F1​=(−10,3) , F2=(−4,1)\mathbf { F } _ { 2 } = ( - 4,1 )F2​=(−4,1) And F3=⟨4,−10⟩\mathbf { F } _ { 3 } = \langle 4 , - 10 \rangleF3​=⟨4,−10⟩ Are acting on a point ppp , find the resultant force

A plane has normal vector n and passes through the point P. Find an equation for the plane. n=(3,0,−13),P(2,4,9)\mathbf { n } = \left( 3,0 , - \frac { 1 } { 3 } \right) , P ( 2,4,9 )n=(3,0,−31​),P(2,4,9)

Find the component of u\mathbf { u } u along Y\mathbf { Y }Y if u=−12i+2j\mathbf { u } = - \frac { 1 } { 2 } \mathbf { i } + 2 \mathbf { j }u=−21​i+2j and v=−2i+12j\mathbf { v } = - 2 \mathbf { i } + \frac { 1 } { 2 } \mathbf { j }v=−2i+21​j .

A water tank is in the shape of a sphere of radius 10 feet. The tank is supported on a metal circle 8 feet below the center of the sphere, as shown in the figure. Find the radius of the metal circle.

Find the work done by the force F=5i+25j\mathbf { F } = 5 \mathbf { i } + 25 \mathbf { j }F=5i+25j in moving an object from P(0,0)P ( 0,0 )P(0,0) to Q(1,6)Q ( 1,6 )Q(1,6) .

Determine whether u=23i+2j\mathbf { u } = 2 \sqrt { 3 } \mathbf { i } + 2 \mathbf { j }u=23​i+2j is orthogonal to v=i−3j\mathbf { v } = \mathbf { i } - \sqrt { 3 } \mathbf { j }v=i−3​j . Justify your answer.

Find the area of the parallelogram determined by the given vectors. u=(0,−2,3),v=⟨5,−5,0)\mathbf { u } = ( 0 , - 2,3 ) , \mathbf { v } = \langle 5 , - 5,0 )u=(0,−2,3),v=⟨5,−5,0)

Find the work done by the force F=100i−25j\mathbf { F } = 100 \mathbf { i } - 25 \mathbf { j }F=100i−25j in moving an object from P(−2,4)P ( - 2,4 )P(−2,4) to Q(150,5)Q ( 150,5 )Q(150,5) .

Equations and Graphs

Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions: Right Triangle Approach

Trigonometric Functions: Unit Circle Approach

Conic Sections

Polar Coordinates and Parametric Equations

Vectors in Two and Three Dimensions

Matrices and Determinants

Conic Sections

Sequences and Series

Counting and Probability

Filters

Find the direction of the vector $v = 6 i + 6 \sqrt { 3 } \mathbf { j }$

Find the vector $\mathbf { Y }$ having length $| \mathrm { v } | = 2$ , and direction $\theta = 135 ^ { \circ }$

Given $\mathbf { u } = ( 4 , - 3 )$ and $\mathrm { v } = \langle , 2,2 \rangle$ , calculate $\mathrm { proj } _ { \mathbf { v } } \mathbf { u }$

A man pulls a sled horizontally by exerting $641 b$ on the rope that is tied to its front end. If the rope makes an angle of $45 ^ { \circ }$ With horizontal, find the work done in moving the sled $35$ Ft

If the vector v has initial point P, what is its terminal point? $\mathbf { v } = ( 0,0,1 ) , P ( 1,1 , - 1 )$

Two vectors u and v are given. Find the angle to one decimal place (expressed in degrees) between u and v. $\mathbf { u } = \langle 4,0 , - 2 \rangle , \mathbf { v } = \langle 2,1,0 \rangle$

Find parametric equations for the line that passes through the point P and is parallel to the vector v. $P ( 0,0,0 ) , \quad \mathrm { v } = \langle - 5,2,9 \rangle$

Find parametric equations for the line that passes through the points P and Q. $P ( 2 , - 1 , - 1 ) , \quad Q ( 0,1 , - 3 )$

Find the center and radius of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x + 4 - 8 y = 0$

Find $\mathbf { u} \cdot \mathbf { Y }$ . $\mathbf { u } = \langle 5,0 ) , \mathbf { v } = ( \sqrt { 3 } , 1 )$

A plane has normal vector n and passes through the point P. Find an equation for the plane. $\mathbf { n } = 5 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } , P ( 0,2 , - 3 )$

Given that the forces $\mathbf { F } _ { 1 } = ( - 10,3 )$ , $\mathbf { F } _ { 2 } = ( - 4,1 )$ And $\mathbf { F } _ { 3 } = \langle 4 , - 10 \rangle$ Are acting on a point $p$ , find the resultant force

A plane has normal vector n and passes through the point P. Find an equation for the plane. $\mathbf { n } = \left( 3,0 , - \frac { 1 } { 3 } \right) , P ( 2,4,9 )$

Find the component of $\mathbf { u }$ along $\mathbf { Y }$ if $\mathbf { u } = - \frac { 1 } { 2 } \mathbf { i } + 2 \mathbf { j }$ and $\mathbf { v } = - 2 \mathbf { i } + \frac { 1 } { 2 } \mathbf { j }$ .

Find the work done by the force $\mathbf { F } = 5 \mathbf { i } + 25 \mathbf { j }$ in moving an object from $P ( 0,0 )$ to $Q ( 1,6 )$ .

Determine whether $\mathbf { u } = 2 \sqrt { 3 } \mathbf { i } + 2 \mathbf { j }$ is orthogonal to $\mathbf { v } = \mathbf { i } - \sqrt { 3 } \mathbf { j }$ . Justify your answer.

Find the area of the parallelogram determined by the given vectors. $\mathbf { u } = ( 0 , - 2,3 ) , \mathbf { v } = \langle 5 , - 5,0 )$

Find the work done by the force $\mathbf { F } = 100 \mathbf { i } - 25 \mathbf { j }$ in moving an object from $P ( - 2,4 )$ to $Q ( 150,5 )$ .