Question 1

Find the distance between the points $P ( 5,0,10 )$ 
and $Q ( 12 , - 3,5 )$ 
.

Accepted Answer

The answer of Find the distance between the points \(P...

Question 2

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, magnitude and the additional force required in order for the forces to be in equilibrium.

Accepted Answer

The resultant force

Question 3

Find $\mathbf { u }  \cdot \mathbf { Y }$ 
. $\mathbf { u } = \langle - 3,2 \rangle , \mathbf { v } = \langle 6 , - 2 \rangle$

Accepted Answer

The answer of Find \(\mathbf { u }  \cdot...

Question 4

Find an equation of a sphere with the given radius r and center C. $\gamma = \sqrt { 7 }$ 
; $C ( 3 , - 1,0 )$

Accepted Answer

The answer of Find an equation of a sphere with...

Question 5

The lengths of two vectors a and b, and the angle F1F1F1S1F1F1F10 between them, are given. Find the length of their cross product, $| \mathbf { a } \times \mathbf { b } |$ 
. Write the answer correct to three decimal places. $| \mathbf { a } | = 0.12 , \mathbf { b } \mid = 1.25 , \theta = 85 ^ { \circ }$

Accepted Answer

The answer of The lengths of two vectors a and...

Question 6

$\mathbf { u } = \left( - \frac { 1 } { 2 } , 2 \right\rangle$ is orthogonal to $\mathbf { v } = \left( - 2 , \frac { 1 } { 2 } \right)$

Accepted Answer

A) True 
 B)False

Question 7

Find an equation of a sphere with the given radius r and center C. $\gamma = \sqrt { 10 }$ 
; $C ( - 10,0,1 )$

Accepted Answer

The answer of Find an equation of a sphere with...

Question 8

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 9

Express the given vector in terms of the unit vectors i, j, and k. $\left( - a , - \frac { 1 } { 2 } a , 4 \right)$

Accepted Answer

The answer of Express the given vector in terms of...

Question 10

Find the distance between the points $P ( 5,0,10 )$ and $Q ( 12 , - 3,5 )$

Accepted Answer

A) 83 
B)  $\sqrt { 83 }$ 
C)  $2 \sqrt { 2 }$ 
D)  $81$ 
E)  9 
A) 83 
B)  $\sqrt { 83 }$ 
C)  $2 \sqrt { 2 }$ 
D)  $81$ 
E)  9

Question 11

If the vector v has initial point P, what is its terminal point? $v = ( 23 , - 5,13 ) , P ( - 6,5,2 )$

Accepted Answer

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

Question 12

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

A)  $6 \mathrm { ft }$ 
B)  $2 \mathrm { ft }$ 
C)  $3 \mathrm { ft }$ 
D)  $5 \mathrm { ft }$ 
E)  none of these 
A)  $6 \mathrm { ft }$ 
B)  $2 \mathrm { ft }$ 
C)  $3 \mathrm { ft }$ 
D)  $5 \mathrm { ft }$ 
E)  none of these

Question 13

Find $\mathbf { u }  \cdot \mathbf { Y }$ . $\mathbf { u } = \langle - 3,2 \rangle , \mathbf { v } = \langle 6 , - 2 \rangle$

Accepted Answer

A)  $22$ 
B)  $13$ 
C)  $- 22$ 
D)  $14$ 
E)  $- 14$ 
A)  $22$ 
B)  $13$ 
C)  $- 22$ 
D)  $14$ 
E)  $- 14$

Question 14

Find $| \mathrm { v } |$ and $| \mathbf { u } + \mathbf { y } |$
, given that $\mathbf { u } = - \mathbf { j }$
And $\mathbf { v } = \mathbf { i }$

Accepted Answer

A)  $| \mathrm { v } | = 1$
, $| \mathbf { u } + \mathrm { v } | = \sqrt { 2 }$ 
B)  $| v | = - 1$
, $| \mathbf { u } + \mathbf { v } | = - \sqrt { 2 }$ 
C)  $| \mathrm { v } | = \sqrt { 2 }$
, $| \mathbf { u } + \mathbf { y } | = 1$ 
D)  $| \mathrm { v } | = \sqrt { 2 }$
, $| \mathbf { u } + \mathrm { v } | = \sqrt { 2 }$ 
E) none 
A)  $| \mathrm { v } | = 1$
, $| \mathbf { u } + \mathrm { v } | = \sqrt { 2 }$ 
B)  $| v | = - 1$
, $| \mathbf { u } + \mathbf { v } | = - \sqrt { 2 }$ 
C)  $| \mathrm { v } | = \sqrt { 2 }$
, $| \mathbf { u } + \mathbf { y } | = 1$ 
D)  $| \mathrm { v } | = \sqrt { 2 }$
, $| \mathbf { u } + \mathrm { v } | = \sqrt { 2 }$ 
E) none

Question 15

A man pulls a sled horizontally by exerting $321 \mathrm {~b}$ 
on the rope that's tied to its front end. If the rope makes an angle of $45 ^ { \circ }$ 
with horizontal, find the work done in moving the sled $55$ 
ft.

Accepted Answer

@#LAT-DLM& 
, @#LAT-DLM& 
. Th

Question 16

Given $\mathbf { u } = \langle 10,5 \rangle$ 
and $v = ( 3,4 )$ 
, calculate $\mathrm { proj } _ { \mathbf { v } } \mathbf { u }$ 
and then resolve $u$ 
into $\mathbf { u } _ { 1 }$ 
and $\mathbf { u } _ { 2 }$ 
.

Accepted Answer

The answer of Given \(\mathbf { u } = \langle...

Question 17

Find the component of $u$ along $\mathbf { Y }$
If $\mathbf { u } = ( - 6,7 )$ and $v = ( 3,4 )$

Accepted Answer

A)  $2$ 
B)  $5$ 
C)  $\frac { \sqrt { 10 } } { 7 }$ 
D)  $\frac { \sqrt { 5 } } { 2 }$ 
E)  none 
A)  $2$ 
B)  $5$ 
C)  $\frac { \sqrt { 10 } } { 7 }$ 
D)  $\frac { \sqrt { 5 } } { 2 }$ 
E)  none

Question 18

The lengths of two vectors a and b, and the angle F1F1F1S1F1F1F10 between them, are given. Find the length of their cross product, $| \mathbf { a } \times \mathbf { b } |$ 
. $| \mathbf { a } | = 4 , \mathbf { b } \mid = 5 , \theta = 60 ^ { \circ }$

Accepted Answer

The answer of The lengths of two vectors a and...

Question 19

Find a vector that is perpendicular to the plane passing through the three given points. $P ( 3,4,5 ) , Q ( 1,0,3 ) , R ( 4,7,6 )$

Accepted Answer

The answer of Find a vector that is perpendicular to...

Question 20

Find an equation of the plane that passes through the points P, Q, and R. $P ( 8,1,1 ) , Q ( 5,2,0 ) , R ( 0,0,0 )$

Accepted Answer

A)  $2 x - 5 y - 11 z = 0$ 
B)  $x - 5 y - z = 0$ 
C)  $2 x - 5 y = 11$ 
D)  $8 x - y - z = 0$ 
E)  none of these 
A)  $2 x - 5 y - 11 z = 0$ 
B)  $x - 5 y - z = 0$ 
C)  $2 x - 5 y = 11$ 
D)  $8 x - y - z = 0$ 
E)  none of these

Find the distance between the points $P ( 5,0,10 )$ and $Q ( 12 , - 3,5 )$ .

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, magnitude and the additional force required in order for the forces to be in equilibrium.

Find $\mathbf { u } \cdot \mathbf { Y }$ . $\mathbf { u } = \langle - 3,2 \rangle , \mathbf { v } = \langle 6 , - 2 \rangle$

Find an equation of a sphere with the given radius r and center C. $\gamma = \sqrt { 7 }$ ; $C ( 3 , - 1,0 )$

The lengths of two vectors a and b, and the angle  between them, are given. Find the length of their cross product, $| \mathbf { a } \times \mathbf { b } |$ . Write the answer correct to three decimal places. $| \mathbf { a } | = 0.12 , \mathbf { b } \mid = 1.25 , \theta = 85 ^ { \circ }$

$\mathbf { u } = \left( - \frac { 1 } { 2 } , 2 \right\rangle$ is orthogonal to $\mathbf { v } = \left( - 2 , \frac { 1 } { 2 } \right)$

Find an equation of a sphere with the given radius r and center C. $\gamma = \sqrt { 10 }$ ; $C ( - 10,0,1 )$

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

Express the given vector in terms of the unit vectors i, j, and k. $\left( - a , - \frac { 1 } { 2 } a , 4 \right)$

Find the distance between the points $P ( 5,0,10 )$ and $Q ( 12 , - 3,5 )$

If the vector v has initial point P, what is its terminal point? $v = ( 23 , - 5,13 ) , P ( - 6,5,2 )$

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 $\mathbf { u } \cdot \mathbf { Y }$ . $\mathbf { u } = \langle - 3,2 \rangle , \mathbf { v } = \langle 6 , - 2 \rangle$

Find $| \mathrm { v } |$ and $| \mathbf { u } + \mathbf { y } |$ , given that $\mathbf { u } = - \mathbf { j }$ And $\mathbf { v } = \mathbf { i }$

A man pulls a sled horizontally by exerting $321 \mathrm {~b}$ on the rope that's tied to its front end. If the rope makes an angle of $45 ^ { \circ }$ with horizontal, find the work done in moving the sled $55$ ft.

Given $\mathbf { u } = \langle 10,5 \rangle$ and $v = ( 3,4 )$ , calculate $\mathrm { proj } _ { \mathbf { v } } \mathbf { u }$ and then resolve $u$ into $\mathbf { u } _ { 1 }$ and $\mathbf { u } _ { 2 }$ .

Find the component of $u$ along $\mathbf { Y }$ If $\mathbf { u } = ( - 6,7 )$ and $v = ( 3,4 )$

The lengths of two vectors a and b, and the angle  between them, are given. Find the length of their cross product, $| \mathbf { a } \times \mathbf { b } |$ . $| \mathbf { a } | = 4 , \mathbf { b } \mid = 5 , \theta = 60 ^ { \circ }$

Find a vector that is perpendicular to the plane passing through the three given points. $P ( 3,4,5 ) , Q ( 1,0,3 ) , R ( 4,7,6 )$

Find an equation of the plane that passes through the points P, Q, and R. $P ( 8,1,1 ) , Q ( 5,2,0 ) , R ( 0,0,0 )$

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

Exam 10: Systems of Equations and Inequalities

Find the distance between the points P(5,0,10)P ( 5,0,10 )P(5,0,10) and Q(12,−3,5)Q ( 12 , - 3,5 )Q(12,−3,5) .

Find u⋅Y\mathbf { u } \cdot \mathbf { Y }u⋅Y . u=⟨−3,2⟩,v=⟨6,−2⟩\mathbf { u } = \langle - 3,2 \rangle , \mathbf { v } = \langle 6 , - 2 \rangleu=⟨−3,2⟩,v=⟨6,−2⟩

Find an equation of a sphere with the given radius r and center C. γ=7\gamma = \sqrt { 7 }γ=7​ ; C(3,−1,0)C ( 3 , - 1,0 )C(3,−1,0)

u=(−12,2⟩\mathbf { u } = \left( - \frac { 1 } { 2 } , 2 \right\rangleu=(−21​,2⟩ is orthogonal to v=(−2,12)\mathbf { v } = \left( - 2 , \frac { 1 } { 2 } \right)v=(−2,21​)

Find an equation of a sphere with the given radius r and center C. γ=10\gamma = \sqrt { 10 }γ=10​ ; C(−10,0,1)C ( - 10,0,1 )C(−10,0,1)

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)

Express the given vector in terms of the unit vectors i, j, and k. (−a,−12a,4)\left( - a , - \frac { 1 } { 2 } a , 4 \right)(−a,−21​a,4)

Find the distance between the points P(5,0,10)P ( 5,0,10 )P(5,0,10) and Q(12,−3,5)Q ( 12 , - 3,5 )Q(12,−3,5)

If the vector v has initial point P, what is its terminal point? v=(23,−5,13),P(−6,5,2)v = ( 23 , - 5,13 ) , P ( - 6,5,2 )v=(23,−5,13),P(−6,5,2)

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 u⋅Y\mathbf { u } \cdot \mathbf { Y }u⋅Y . u=⟨−3,2⟩,v=⟨6,−2⟩\mathbf { u } = \langle - 3,2 \rangle , \mathbf { v } = \langle 6 , - 2 \rangleu=⟨−3,2⟩,v=⟨6,−2⟩

Find ∣v∣| \mathrm { v } |∣v∣ and ∣u+y∣| \mathbf { u } + \mathbf { y } |∣u+y∣ , given that u=−j\mathbf { u } = - \mathbf { j }u=−j And v=i\mathbf { v } = \mathbf { i }v=i

A man pulls a sled horizontally by exerting 321 b321 \mathrm {~b}321 b on the rope that's 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 555555 ft.

Given u=⟨10,5⟩\mathbf { u } = \langle 10,5 \rangleu=⟨10,5⟩ and v=(3,4)v = ( 3,4 )v=(3,4) , calculate projvu\mathrm { proj } _ { \mathbf { v } } \mathbf { u }projv​u and then resolve uuu into u1\mathbf { u } _ { 1 }u1​ and u2\mathbf { u } _ { 2 }u2​ .

Find the component of uuu along Y\mathbf { Y }Y If u=(−6,7)\mathbf { u } = ( - 6,7 )u=(−6,7) and v=(3,4)v = ( 3,4 )v=(3,4)

Find a vector that is perpendicular to the plane passing through the three given points. P(3,4,5),Q(1,0,3),R(4,7,6)P ( 3,4,5 ) , Q ( 1,0,3 ) , R ( 4,7,6 )P(3,4,5),Q(1,0,3),R(4,7,6)

Find an equation of the plane that passes through the points P, Q, and R. P(8,1,1),Q(5,2,0),R(0,0,0)P ( 8,1,1 ) , Q ( 5,2,0 ) , R ( 0,0,0 )P(8,1,1),Q(5,2,0),R(0,0,0)

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 distance between the points $P ( 5,0,10 )$ and $Q ( 12 , - 3,5 )$ .

Find $\mathbf { u } \cdot \mathbf { Y }$ . $\mathbf { u } = \langle - 3,2 \rangle , \mathbf { v } = \langle 6 , - 2 \rangle$

Find an equation of a sphere with the given radius r and center C. $\gamma = \sqrt { 7 }$ ; $C ( 3 , - 1,0 )$

$\mathbf { u } = \left( - \frac { 1 } { 2 } , 2 \right\rangle$ is orthogonal to $\mathbf { v } = \left( - 2 , \frac { 1 } { 2 } \right)$

Find an equation of a sphere with the given radius r and center C. $\gamma = \sqrt { 10 }$ ; $C ( - 10,0,1 )$

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

Express the given vector in terms of the unit vectors i, j, and k. $\left( - a , - \frac { 1 } { 2 } a , 4 \right)$

Find the distance between the points $P ( 5,0,10 )$ and $Q ( 12 , - 3,5 )$

If the vector v has initial point P, what is its terminal point? $v = ( 23 , - 5,13 ) , P ( - 6,5,2 )$

Find $\mathbf { u } \cdot \mathbf { Y }$ . $\mathbf { u } = \langle - 3,2 \rangle , \mathbf { v } = \langle 6 , - 2 \rangle$

Find $| \mathrm { v } |$ and $| \mathbf { u } + \mathbf { y } |$ , given that $\mathbf { u } = - \mathbf { j }$ And $\mathbf { v } = \mathbf { i }$

A man pulls a sled horizontally by exerting $321 \mathrm {~b}$ on the rope that's tied to its front end. If the rope makes an angle of $45 ^ { \circ }$ with horizontal, find the work done in moving the sled $55$ ft.

Given $\mathbf { u } = \langle 10,5 \rangle$ and $v = ( 3,4 )$ , calculate $\mathrm { proj } _ { \mathbf { v } } \mathbf { u }$ and then resolve $u$ into $\mathbf { u } _ { 1 }$ and $\mathbf { u } _ { 2 }$ .

Find the component of $u$ along $\mathbf { Y }$ If $\mathbf { u } = ( - 6,7 )$ and $v = ( 3,4 )$

Find a vector that is perpendicular to the plane passing through the three given points. $P ( 3,4,5 ) , Q ( 1,0,3 ) , R ( 4,7,6 )$

Find an equation of the plane that passes through the points P, Q, and R. $P ( 8,1,1 ) , Q ( 5,2,0 ) , R ( 0,0,0 )$