Question 1

Which of the following is not a quadric surface?

Accepted Answer

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

Question 2

Find an equation of the surface consisting of all points that are equidistant from the y-axis
and the xz-plane: Then sketch the surface.

Accepted Answer

The answer of Find an equation of the surface consisting...

Question 3

Find the direction vector of the line which is the intersection of the planes x + y + z = 1 and x + y - z = 2.

Accepted Answer

A)  $\{ 2,2 , - 2 \}$ 
B)  $\{ 0 , - 2,2 \}$ 
C)  $\{ - 2,2,0 \}$ 
D)  $\{ 2,2,0 \}$ 
E)  $( 2,2,2 )$ 
F) $\{ 0 , - 2 , - 2 \}$ 
G) $\{ - 2 , - 2,0 \}$ 
H) $\{ 0,2,2 \}$ 
A)  $\{ 2,2 , - 2 \}$ 
B)  $\{ 0 , - 2,2 \}$ 
C)  $\{ - 2,2,0 \}$ 
D)  $\{ 2,2,0 \}$ 
E)  $( 2,2,2 )$ 
F) $\{ 0 , - 2 , - 2 \}$ 
G) $\{ - 2 , - 2,0 \}$ 
H) $\{ 0,2,2 \}$

Question 4

If $\mathbf { b } = \langle 1,0,1 \rangle$ , $\mathbf { c } = \langle 0 , - 1,1 \rangle$ , and $\mathbf { a } = \langle 2 , - 3 , z \rangle$ , find a value for z which guarantees that a, b, and c are coplanar.

Accepted Answer

The answer of If \(\mathbf { b } = \langle...

Question 5

Find the set of intersection of the surfaces whose equations in spherical coordinates are $\phi = \frac { \pi } { 4 }$ and $\theta = 0$ .

Accepted Answer

A ray z =

Question 6

Find the projection of the vector $\langle 2,3 , - 5 \rangle$ along the vector $\langle - 1,1,2 \rangle$ .

Accepted Answer

The answer of Find the projection of the vector \(\langle...

Question 7

Describe the trace of the surface z = $4 x ^ { 2 }$ = 0 in the plane z = 1.

Accepted Answer

The answer of Describe the trace of the surface z...

Question 8

Describe in words the solid represented in spherical coordinates by the inequality $2 \leq \rho \leq 5$ .

Accepted Answer

solid sphere of radi

Question 9

Find equations of all planes containing the x-axis.

Accepted Answer

by + cz =

Question 10

Sketch the curve of intersection of the two surfaces $z = y ^ { 2 } - x ^ { 2 }$ and $x ^ { 2 } + y ^ { 2 } = 1$ .

Accepted Answer

The answer of Sketch the curve of intersection of the...

Question 11

Find the volume of the parallelepiped determined by the vectors $\{ 1,0,1 \}$ , $\{ 2,1,3 \}$ , and $\{ 1,1,1 \}$ .

Accepted Answer

A) 1 
B) 2 
C) 3 
D) 4 
E) 5
F)6
G)7
H)8 
A) 1 
B) 2 
C) 3 
D) 4 
E) 5
F)6
G)7
H)8

Question 12

Describe the surface whose equation in cylindrical coordinates is $r = \sin \theta$ .

Accepted Answer

Circular c

Question 13

Convert $\left( 2 , \frac { 5 \pi } { 4 } , 3 \right)$ from cylindrical coordinates to rectangular coordinates.

Accepted Answer

A)  $( 1,1,3 )$ 
B)  $( 0,2,3 )$ 
C)  $( 2,0,3 )$ 
D)  $( \sqrt { 2 } , \sqrt { 2 } , 3 )$ 
E)  $( - 1 , - 1,3 )$ 
F) $( 0 , - 2,3 )$ 
G) $( - 2,0,3 )$ 
H) $( - \sqrt { 2 } , - \sqrt { 2 } , 3 )$ 
A)  $( 1,1,3 )$ 
B)  $( 0,2,3 )$ 
C)  $( 2,0,3 )$ 
D)  $( \sqrt { 2 } , \sqrt { 2 } , 3 )$ 
E)  $( - 1 , - 1,3 )$ 
F) $( 0 , - 2,3 )$ 
G) $( - 2,0,3 )$ 
H) $( - \sqrt { 2 } , - \sqrt { 2 } , 3 )$

Question 14

Given the points P $( - 2,5 )$ and Q $( - 3,3 )$ , find the unit vector in the direction of the displacement vector $\overrightarrow { P Q }$ .

Accepted Answer

The answer of Given the points P \(( - 2,5...

Question 15

Find the length of the vector $\langle 2,2,3 \rangle$ .

Accepted Answer

A)  $\sqrt { 13 }$ 
B)  $\sqrt { 14 }$ 
C)  $\sqrt { 15 }$ 
D) 4 
E)  $\sqrt { 17 }$ 
F) $\sqrt { 18 }$ 
G) $\sqrt { 19 }$ 
H) $\sqrt { 20 }$ 
A)  $\sqrt { 13 }$ 
B)  $\sqrt { 14 }$ 
C)  $\sqrt { 15 }$ 
D) 4 
E)  $\sqrt { 17 }$ 
F) $\sqrt { 18 }$ 
G) $\sqrt { 19 }$ 
H) $\sqrt { 20 }$

Question 16

Find symmetric equations of the line passing through (2, -3, 4) and parallel to the vector AB, where A and B are the points (-2, 1, 1) and (0, 2, 3).

Accepted Answer

The answer of Find symmetric equations of the line passing...

Question 17

Sketch and identify the quadric surface given by $z ^ { 2 } - 2 z - y = 0$ .

Accepted Answer

The answer of Sketch and identify the quadric surface given...

Question 18

Let f(x, y) = $\sqrt { x ^ { 2 } + y ^ { 2 } }$ (a) Evaluate f (-3, 0).(b) Find the domain of f.(c) Find the range of f.

Accepted Answer

(a) 3
(b) x and y ca

Question 19

Describe the surface whose equation in cylindrical coordinates is $\phi = \pi$ .

Accepted Answer

A) Cylinder with vertical axis 
B) Cylinder with horizontal axis 
C) Sphere 
D) Positive z-axis 
E) Negative z-axis
F)Paraboloid
G)Cone or half-cone with vertical axis
H)Cone or half-cone with horizontal axis 
A) Cylinder with vertical axis 
B) Cylinder with horizontal axis 
C) Sphere 
D) Positive z-axis 
E) Negative z-axis
F)Paraboloid
G)Cone or half-cone with vertical axis
H)Cone or half-cone with horizontal axis

Question 20

Sketch and identify the quadric surface given by $x ^ { 2 } + y ^ { 2 } - 2 x = 0$ .

Accepted Answer

The answer of Sketch and identify the quadric surface given...

Which of the following is not a quadric surface?

Find an equation of the surface consisting of all points that are equidistant from the y-axis and the xz-plane: Then sketch the surface.

Find the direction vector of the line which is the intersection of the planes x + y + z = 1 and x + y - z = 2.

If $\mathbf { b } = \langle 1,0,1 \rangle$ , $\mathbf { c } = \langle 0 , - 1,1 \rangle$ , and $\mathbf { a } = \langle 2 , - 3 , z \rangle$ , find a value for z which guarantees that a, b, and c are coplanar.

Find the set of intersection of the surfaces whose equations in spherical coordinates are $\phi = \frac { \pi } { 4 }$ and $\theta = 0$ .

Find the projection of the vector $\langle 2,3 , - 5 \rangle$ along the vector $\langle - 1,1,2 \rangle$ .

Describe the trace of the surface z = $4 x ^ { 2 }$ = 0 in the plane z = 1.

Describe in words the solid represented in spherical coordinates by the inequality $2 \leq \rho \leq 5$ .

Find equations of all planes containing the x-axis.

Sketch the curve of intersection of the two surfaces $z = y ^ { 2 } - x ^ { 2 }$ and $x ^ { 2 } + y ^ { 2 } = 1$ . $Sketch the curve of intersection of the two surfaces z = y ^ { 2 } - x ^ { 2 } and x ^ { 2 } + y ^ { 2 } = 1 .$

Find the volume of the parallelepiped determined by the vectors $\{ 1,0,1 \}$ , $\{ 2,1,3 \}$ , and $\{ 1,1,1 \}$ .

Describe the surface whose equation in cylindrical coordinates is $r = \sin \theta$ .

Convert $\left( 2 , \frac { 5 \pi } { 4 } , 3 \right)$ from cylindrical coordinates to rectangular coordinates.

Given the points P $( - 2,5 )$ and Q $( - 3,3 )$ , find the unit vector in the direction of the displacement vector $\overrightarrow { P Q }$ .

Find the length of the vector $\langle 2,2,3 \rangle$ .

Find symmetric equations of the line passing through (2, -3, 4) and parallel to the vector AB, where A and B are the points (-2, 1, 1) and (0, 2, 3).

Sketch and identify the quadric surface given by $z ^ { 2 } - 2 z - y = 0$ . $Sketch and identify the quadric surface given by z ^ { 2 } - 2 z - y = 0 .$

Let f(x, y) = $\sqrt { x ^ { 2 } + y ^ { 2 } }$ (a) Evaluate f (-3, 0).(b) Find the domain of f.(c) Find the range of f.

Describe the surface whose equation in cylindrical coordinates is $\phi = \pi$ .

Sketch and identify the quadric surface given by $x ^ { 2 } + y ^ { 2 } - 2 x = 0$ . $Sketch and identify the quadric surface given by x ^ { 2 } + y ^ { 2 } - 2 x = 0 .$

Functions and Models

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Integrals

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Exam 9: Vectors and the Geometry of Space

Which of the following is not a quadric surface?

Find an equation of the surface consisting of all points that are equidistant from the y-axis and the xz-plane: Then sketch the surface.

Find the direction vector of the line which is the intersection of the planes x + y + z = 1 and x + y - z = 2.

If b=⟨1,0,1⟩\mathbf { b } = \langle 1,0,1 \rangleb=⟨1,0,1⟩ , c=⟨0,−1,1⟩\mathbf { c } = \langle 0 , - 1,1 \ranglec=⟨0,−1,1⟩ , and a=⟨2,−3,z⟩\mathbf { a } = \langle 2 , - 3 , z \ranglea=⟨2,−3,z⟩ , find a value for z which guarantees that a, b, and c are coplanar.

Find the set of intersection of the surfaces whose equations in spherical coordinates are ϕ=π4\phi = \frac { \pi } { 4 }ϕ=4π​ and θ=0\theta = 0θ=0 .

Find the projection of the vector ⟨2,3,−5⟩\langle 2,3 , - 5 \rangle⟨2,3,−5⟩ along the vector ⟨−1,1,2⟩\langle - 1,1,2 \rangle⟨−1,1,2⟩ .

Describe the trace of the surface z = 4x24 x ^ { 2 }4x2 = 0 in the plane z = 1.

Describe in words the solid represented in spherical coordinates by the inequality 2≤ρ≤52 \leq \rho \leq 52≤ρ≤5 .

Find equations of all planes containing the x-axis.

Sketch the curve of intersection of the two surfaces z=y2−x2z = y ^ { 2 } - x ^ { 2 }z=y2−x2 and x2+y2=1x ^ { 2 } + y ^ { 2 } = 1x2+y2=1 .

Find the volume of the parallelepiped determined by the vectors {1,0,1}\{ 1,0,1 \}{1,0,1} , {2,1,3}\{ 2,1,3 \}{2,1,3} , and {1,1,1}\{ 1,1,1 \}{1,1,1} .

Describe the surface whose equation in cylindrical coordinates is r=sin⁡θr = \sin \thetar=sinθ .

Convert (2,5π4,3)\left( 2 , \frac { 5 \pi } { 4 } , 3 \right)(2,45π​,3) from cylindrical coordinates to rectangular coordinates.

Given the points P (−2,5)( - 2,5 )(−2,5) and Q (−3,3)( - 3,3 )(−3,3) , find the unit vector in the direction of the displacement vector PQ→\overrightarrow { P Q }PQ​ .

Find the length of the vector ⟨2,2,3⟩\langle 2,2,3 \rangle⟨2,2,3⟩ .

Find symmetric equations of the line passing through (2, -3, 4) and parallel to the vector AB, where A and B are the points (-2, 1, 1) and (0, 2, 3).

Sketch and identify the quadric surface given by z2−2z−y=0z ^ { 2 } - 2 z - y = 0z2−2z−y=0 .

Let f(x, y) = x2+y2\sqrt { x ^ { 2 } + y ^ { 2 } }x2+y2​ (a) Evaluate f (-3, 0).(b) Find the domain of f.(c) Find the range of f.

Describe the surface whose equation in cylindrical coordinates is ϕ=π\phi = \piϕ=π .

Sketch and identify the quadric surface given by x2+y2−2x=0x ^ { 2 } + y ^ { 2 } - 2 x = 0x2+y2−2x=0 .

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Differential Equations

Infinite Sequences and Series

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Filters

If $\mathbf { b } = \langle 1,0,1 \rangle$ , $\mathbf { c } = \langle 0 , - 1,1 \rangle$ , and $\mathbf { a } = \langle 2 , - 3 , z \rangle$ , find a value for z which guarantees that a, b, and c are coplanar.

Find the set of intersection of the surfaces whose equations in spherical coordinates are $\phi = \frac { \pi } { 4 }$ and $\theta = 0$ .

Find the projection of the vector $\langle 2,3 , - 5 \rangle$ along the vector $\langle - 1,1,2 \rangle$ .

Describe the trace of the surface z = $4 x ^ { 2 }$ = 0 in the plane z = 1.

Describe in words the solid represented in spherical coordinates by the inequality $2 \leq \rho \leq 5$ .

Sketch the curve of intersection of the two surfaces $z = y ^ { 2 } - x ^ { 2 }$ and $x ^ { 2 } + y ^ { 2 } = 1$ . $Sketch the curve of intersection of the two surfaces z = y ^ { 2 } - x ^ { 2 } and x ^ { 2 } + y ^ { 2 } = 1 .$

Find the volume of the parallelepiped determined by the vectors $\{ 1,0,1 \}$ , $\{ 2,1,3 \}$ , and $\{ 1,1,1 \}$ .

Describe the surface whose equation in cylindrical coordinates is $r = \sin \theta$ .

Convert $\left( 2 , \frac { 5 \pi } { 4 } , 3 \right)$ from cylindrical coordinates to rectangular coordinates.

Given the points P $( - 2,5 )$ and Q $( - 3,3 )$ , find the unit vector in the direction of the displacement vector $\overrightarrow { P Q }$ .

Find the length of the vector $\langle 2,2,3 \rangle$ .

Sketch and identify the quadric surface given by $z ^ { 2 } - 2 z - y = 0$ . $Sketch and identify the quadric surface given by z ^ { 2 } - 2 z - y = 0 .$

Let f(x, y) = $\sqrt { x ^ { 2 } + y ^ { 2 } }$ (a) Evaluate f (-3, 0).(b) Find the domain of f.(c) Find the range of f.

Describe the surface whose equation in cylindrical coordinates is $\phi = \pi$ .

Sketch and identify the quadric surface given by $x ^ { 2 } + y ^ { 2 } - 2 x = 0$ . $Sketch and identify the quadric surface given by x ^ { 2 } + y ^ { 2 } - 2 x = 0 .$