Question 1

Find a vector that is perpendicular to the plane passing through the three given points. $P ( 3,0,1 ) , Q ( 0,2 , - 5 ) , R ( - 2,0,5 )$

Accepted Answer

$\langle 8,42,10 \rangle$

Question 2

Find the vectors $\mathbf { u } + \mathbf { v }$ , $\mathbf { u } - \mathbf { v }$ , and $3 \mathbf { u } - \frac { 1 } { 2 } \mathbf { v }$ . $\mathbf { u } = \langle a , 2 b , 3 c \rangle , \mathbf { v } = \langle - a , b , - 2 c \rangle$

Accepted Answer

$\mathbf { u } + \mathbf { v } = \langle 0,3 b , c \rangle ; \mathbf { u } - \mathbf { v } = \langle 2 a , b , 5c \rangle ; 3 \mathbf { u } - \frac { 1 } { 2 } \mathbf { v } = \left\langle \frac { 7 } { 2 } a , \frac { 11 } { 2 } b , 10 c \right\rangle$

Question 3

Determine whether $\mathbf { u } = \left\langle - \frac { 1 } { 2 } , 2 \right\rangle$ is orthogonal to $\mathbf { v } = \left\langle - 2 , \frac { 1 } { 2 } \right\rangle$ .

Accepted Answer

$\mathbf { u } \cdot \mathbf { v } = \left( - \frac { 1 } { 2 } \right) ( - 2 ) + ( 2 ) \left( \frac { 1 } { 2 } \right) = 2$ , not orthogonal

Question 4

Given $\mathbf { u } = \langle 4 , - 3 \rangle$ and $\mathbf { v } = \langle 9,2 \rangle$ , calculate $\operatorname { proj } _ { \mathbf { v } } \mathbf { u }$ and then resolve $\mathbf { u }$ into $\mathbf { u } _ { 1 }$ and $\mathbf { u } _ { 2 }$ .

Accepted Answer

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

Question 5

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

Accepted Answer

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

Question 6

A description of a line is given. Find parametric equations for the line.
The line perpendicular to the xz-plane that contains the point $( 3 , - 1,4 )$ .

Accepted Answer

The answer of A description of a line is given....

Question 7

Describe the surface represented by the given equation. $x = 3$

Accepted Answer

Plane para

Question 8

The lengths of two vectors a and b, and the angle $ \theta $ between them, are given. Find the length of their cross product, $| \mathbf { a } \times \mathbf { b } |$ . $| \mathbf { a } | = 0.12 , | \mathbf { b } | = 1.25 , \theta = 85 ^ { \circ }$

Accepted Answer

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

Question 9

Find $| v |$ and $| \mathbf { u } + \mathbf { v } |$ , given that $\mathbf { u } = 2 \mathbf { i } - \mathbf { j }$ and $\mathbf { v } = \mathbf { i }$ .

Accepted Answer

The answer of Find $| v |$ and \(| \mathbf...

Question 10

Two vectors u and v are given. Find the angle (expressed in degrees) between u and v. $\mathbf { u } = \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } , \mathbf { v } = \mathbf { j } + \mathbf { k }$

Accepted Answer

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

Question 11

Find the vector v with initial point P and terminal point Q. $P ( 2 , - 1,0 )$ , $Q ( 0 , - 2,5 )$

Accepted Answer

The answer of Find the vector v with initial point...

Question 12

Two vectors u and v are given. Find their dot product $\mathbf { u } \cdot \mathbf { v }$ . $\mathbf { u } = \langle - 3,0,3  \rangle , \mathbf { v } = \langle 2,4 , \frac { 1 } { 3 }  \rangle$

Accepted Answer

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

Question 13

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

Accepted Answer

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

Question 14

Given that the forces $\mathbf { F } _ { 1 } = \langle - 10,3 \rangle$ , $\mathbf { F } _ { 2 } = \langle - 4,1 \rangle$ and $$\mathbf { F } _ { ] } = \langle 4 , - 10 \rangle$$ 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 15

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 16

Describe the trace of the sphere $( x - 1 ) ^ { 2 } + ( y - 4 ) ^ { 2 } + ( z - 3 ) ^ { 2 } = 144$ through 
(a) the xz-plane in
(b) the plane
 $z = - 2$ .

Accepted Answer

a) circle

Question 17

Find $\left| \frac { 1 } { 2 } \mathbf { v } \right|$ , $| \mathbf { u } - \mathbf { v } |$ and $| \mathbf { u } | - | \mathbf { v } |$ , given that $\mathbf { u } = \langle 1$ , $\text { 3) }$ and $\mathbf { v } = \langle - 2,1 \rangle$ .

Accepted Answer

The answer of Find \(\left| \frac { 1 } {...

Question 18

Given $\mathbf { u }$ and $\mathbf { v }$ in the figure, sketch $\mathbf { v } - \mathbf { u }$ .

Accepted Answer

The answer of Given $\mathbf { u }$ and \(\mathbf...

Question 19

Find the vector with initial point $P ( 6,3 )$ and terminal point $Q ( - 6 , - 3 )$ .

Accepted Answer

The vector

Question 20

Find the area of the parallelogram determined by the given vectors. $\mathbf { u } = 2 \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } , \mathbf { v } = 2 \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k }$

Accepted Answer

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

Find a vector that is perpendicular to the plane passing through the three given points. $P ( 3,0,1 ) , Q ( 0,2 , - 5 ) , R ( - 2,0,5 )$

Find the vectors $\mathbf { u } + \mathbf { v }$ , $\mathbf { u } - \mathbf { v }$ , and $3 \mathbf { u } - \frac { 1 } { 2 } \mathbf { v }$ . $\mathbf { u } = \langle a , 2 b , 3 c \rangle , \mathbf { v } = \langle - a , b , - 2 c \rangle$

Determine whether $\mathbf { u } = \left\langle - \frac { 1 } { 2 } , 2 \right\rangle$ is orthogonal to $\mathbf { v } = \left\langle - 2 , \frac { 1 } { 2 } \right\rangle$ .

Given $\mathbf { u } = \langle 4 , - 3 \rangle$ and $\mathbf { v } = \langle 9,2 \rangle$ , calculate $\operatorname { proj } _ { \mathbf { v } } \mathbf { u }$ and then resolve $\mathbf { u }$ into $\mathbf { u } _ { 1 }$ and $\mathbf { u } _ { 2 }$ .

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

A description of a line is given. Find parametric equations for the line. The line perpendicular to the xz-plane that contains the point $( 3 , - 1,4 )$ .

Describe the surface represented by the given equation. $x = 3$

The lengths of two vectors a and b, and the angle $\theta$ between them, are given. Find the length of their cross product, $| \mathbf { a } \times \mathbf { b } |$ . $| \mathbf { a } | = 0.12 , | \mathbf { b } | = 1.25 , \theta = 85 ^ { \circ }$

Find $| v |$ and $| \mathbf { u } + \mathbf { v } |$ , given that $\mathbf { u } = 2 \mathbf { i } - \mathbf { j }$ and $\mathbf { v } = \mathbf { i }$ .

Two vectors u and v are given. Find the angle (expressed in degrees) between u and v. $\mathbf { u } = \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } , \mathbf { v } = \mathbf { j } + \mathbf { k }$

Find the vector v with initial point P and terminal point Q. $P ( 2 , - 1,0 )$ , $Q ( 0 , - 2,5 )$

Two vectors u and v are given. Find their dot product $\mathbf { u } \cdot \mathbf { v }$ . $\mathbf { u } = \langle - 3,0,3 \rangle , \mathbf { v } = \langle 2,4 , \frac { 1 } { 3 } \rangle$

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

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

Describe the trace of the sphere $( x - 1 ) ^ { 2 } + ( y - 4 ) ^ { 2 } + ( z - 3 ) ^ { 2 } = 144$ through (a) the xz-plane in (b) the plane $z = - 2$ .

Find $\left| \frac { 1 } { 2 } \mathbf { v } \right|$ , $| \mathbf { u } - \mathbf { v } |$ and $| \mathbf { u } | - | \mathbf { v } |$ , given that $\mathbf { u } = \langle 1$ , $\text { 3) }$ and $\mathbf { v } = \langle - 2,1 \rangle$ .

Given $\mathbf { u }$ and $\mathbf { v }$ in the figure, sketch $\mathbf { v } - \mathbf { u }$ . $Given \mathbf { u } and \mathbf { v } in the figure, sketch \mathbf { v } - \mathbf { u } .$

Find the vector with initial point $P ( 6,3 )$ and terminal point $Q ( - 6 , - 3 )$ .

Find the area of the parallelogram determined by the given vectors. $\mathbf { u } = 2 \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } , \mathbf { v } = 2 \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k }$

Fundamentals

Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions: Unit Circle Approach

Trigonometric Functions: Right Triangle Approach

Analytic Trigonometry

Polar Coordinates and Parametric Equations

Systems of Equations and Inequalities

Conic Sections

Sequences and Series

Limits: a Preview of Calculus

Filters

Exam 9: Vectors in Two and Three Dimensions

Find a vector that is perpendicular to the plane passing through the three given points. P(3,0,1),Q(0,2,−5),R(−2,0,5)P ( 3,0,1 ) , Q ( 0,2 , - 5 ) , R ( - 2,0,5 )P(3,0,1),Q(0,2,−5),R(−2,0,5)

Determine whether u=⟨−12,2⟩\mathbf { u } = \left\langle - \frac { 1 } { 2 } , 2 \right\rangleu=⟨−21​,2⟩ is orthogonal to v=⟨−2,12⟩\mathbf { v } = \left\langle - 2 , \frac { 1 } { 2 } \right\ranglev=⟨−2,21​⟩ .

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

A description of a line is given. Find parametric equations for the line. The line perpendicular to the xz-plane that contains the point (3,−1,4)( 3 , - 1,4 )(3,−1,4) .

Describe the surface represented by the given equation. x=3x = 3x=3

Find ∣v∣| v |∣v∣ and ∣u+v∣| \mathbf { u } + \mathbf { v } |∣u+v∣ , given that u=2i−j\mathbf { u } = 2 \mathbf { i } - \mathbf { j }u=2i−j and v=i\mathbf { v } = \mathbf { i }v=i .

Two vectors u and v are given. Find the angle (expressed in degrees) between u and v. u=i+2j−3k,v=j+k\mathbf { u } = \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } , \mathbf { v } = \mathbf { j } + \mathbf { k }u=i+2j−3k,v=j+k

Find the vector v with initial point P and terminal point Q. P(2,−1,0)P ( 2 , - 1,0 )P(2,−1,0) , Q(0,−2,5)Q ( 0 , - 2,5 )Q(0,−2,5)

Two vectors u and v are given. Find their dot product u⋅v\mathbf { u } \cdot \mathbf { v }u⋅v . u=⟨−3,0,3⟩,v=⟨2,4,13⟩\mathbf { u } = \langle - 3,0,3 \rangle , \mathbf { v } = \langle 2,4 , \frac { 1 } { 3 } \rangleu=⟨−3,0,3⟩,v=⟨2,4,31​⟩

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

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)

Describe the trace of the sphere (x−1)2+(y−4)2+(z−3)2=144( x - 1 ) ^ { 2 } + ( y - 4 ) ^ { 2 } + ( z - 3 ) ^ { 2 } = 144(x−1)2+(y−4)2+(z−3)2=144 through (a) the xz-plane in (b) the plane z=−2z = - 2z=−2 .

Given u\mathbf { u }u and v\mathbf { v }v in the figure, sketch v−u\mathbf { v } - \mathbf { u }v−u .

Find the vector with initial point P(6,3)P ( 6,3 )P(6,3) and terminal point Q(−6,−3)Q ( - 6 , - 3 )Q(−6,−3) .

Find the area of the parallelogram determined by the given vectors. u=2i−2j+2k,v=2i+2j−2k\mathbf { u } = 2 \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } , \mathbf { v } = 2 \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k }u=2i−2j+2k,v=2i+2j−2k

Fundamentals

Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions: Unit Circle Approach

Trigonometric Functions: Right Triangle Approach

Analytic Trigonometry

Polar Coordinates and Parametric Equations

Systems of Equations and Inequalities

Conic Sections

Sequences and Series

Limits: a Preview of Calculus

Filters

Find a vector that is perpendicular to the plane passing through the three given points. $P ( 3,0,1 ) , Q ( 0,2 , - 5 ) , R ( - 2,0,5 )$

Determine whether $\mathbf { u } = \left\langle - \frac { 1 } { 2 } , 2 \right\rangle$ is orthogonal to $\mathbf { v } = \left\langle - 2 , \frac { 1 } { 2 } \right\rangle$ .

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

A description of a line is given. Find parametric equations for the line. The line perpendicular to the xz-plane that contains the point $( 3 , - 1,4 )$ .

Describe the surface represented by the given equation. $x = 3$

Find $| v |$ and $| \mathbf { u } + \mathbf { v } |$ , given that $\mathbf { u } = 2 \mathbf { i } - \mathbf { j }$ and $\mathbf { v } = \mathbf { i }$ .

Two vectors u and v are given. Find the angle (expressed in degrees) between u and v. $\mathbf { u } = \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } , \mathbf { v } = \mathbf { j } + \mathbf { k }$

Find the vector v with initial point P and terminal point Q. $P ( 2 , - 1,0 )$ , $Q ( 0 , - 2,5 )$

Two vectors u and v are given. Find their dot product $\mathbf { u } \cdot \mathbf { v }$ . $\mathbf { u } = \langle - 3,0,3 \rangle , \mathbf { v } = \langle 2,4 , \frac { 1 } { 3 } \rangle$

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

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

Describe the trace of the sphere $( x - 1 ) ^ { 2 } + ( y - 4 ) ^ { 2 } + ( z - 3 ) ^ { 2 } = 144$ through (a) the xz-plane in (b) the plane $z = - 2$ .

Given $\mathbf { u }$ and $\mathbf { v }$ in the figure, sketch $\mathbf { v } - \mathbf { u }$ . $Given \mathbf { u } and \mathbf { v } in the figure, sketch \mathbf { v } - \mathbf { u } .$

Find the vector with initial point $P ( 6,3 )$ and terminal point $Q ( - 6 , - 3 )$ .

Find the area of the parallelogram determined by the given vectors. $\mathbf { u } = 2 \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } , \mathbf { v } = 2 \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k }$