Question 1

Find an equation of the surface consisting of all points that are equidistant from the point (0, 1, 0) and the plane y = -1. Then sketch the surface.

Accepted Answer

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

Question 2

What region of $\mathbb { R } ^ { 3 }$ is represented by the equation z = 3?

Accepted Answer

A) line parallel to the xy-plane 
B) line parallel to the xz-plane 
C) line parallel to the yz-plane 
D) plane parallel to the xz-plane 
E) plane parallel to the xy-plane
F)line perpendicular to the xy-plane
G)line perpendicular to the xz-plane
H)line perpendicular to the yz-plane 
A) line parallel to the xy-plane 
B) line parallel to the xz-plane 
C) line parallel to the yz-plane 
D) plane parallel to the xz-plane 
E) plane parallel to the xy-plane
F)line perpendicular to the xy-plane
G)line perpendicular to the xz-plane
H)line perpendicular to the yz-plane

Question 3

Convert $( 1 , \pi , 1 )$ from cylindrical coordinates to rectangular coordinates.

Accepted Answer

A)  $( 1,1,1 )$ 
B)  $( - 1,1,1 )$ 
C)  $( 1 , - 1,1 )$ 
D)  $( 1,1 , - 1 )$ 
E)  $( - 1,0,1 )$ 
F) $( 0 , - 1,1 )$ 
G) $( 1,1 , - 1 )$ 
H) $( 0,1,1 )$ 
A)  $( 1,1,1 )$ 
B)  $( - 1,1,1 )$ 
C)  $( 1 , - 1,1 )$ 
D)  $( 1,1 , - 1 )$ 
E)  $( - 1,0,1 )$ 
F) $( 0 , - 1,1 )$ 
G) $( 1,1 , - 1 )$ 
H) $( 0,1,1 )$

Question 4

Find equations of all planes containing z-axis.

Accepted Answer

ax + by =

Question 5

Find cylindrical and spherical equations for the surface whose equation in rectangular coordinates is x = 2. Describe the surface.

Accepted Answer

Cylindrica

Question 6

Find an equation of a plane containing the point P(2, -1, 1) and the y-axis.

Accepted Answer

A)  $y + z = 0$ 
B)  $x - y = 3$ 
C)  $x = 2$ 
D)  $x - 2 z = 0$ 
E)  $y - z = 0$ 
F) $x + 2 y = 0$ 
G) $z = 1$ 
H) $2 x - z = 3$ 
A)  $y + z = 0$ 
B)  $x - y = 3$ 
C)  $x = 2$ 
D)  $x - 2 z = 0$ 
E)  $y - z = 0$ 
F) $x + 2 y = 0$ 
G) $z = 1$ 
H) $2 x - z = 3$

Question 7

Find rectangular and cylindrical equations for the surface whose equation in spherical coordinates is $\beta = 3$ . Describe the surface.

Accepted Answer

Rectangula

Question 8

Find rectangular and spherical equations for the surface whose equation in cylindrical coordinates is $\theta = \frac { \pi } { 4 }$ . Describe the surface.

Accepted Answer

Rectangula

Question 9

Describe the surface whose equation in spherical coordinates is $\sin \phi = \cos \phi$ .

Accepted Answer

Half-cone

Question 10

Find the distance from the point (4, -3, 5) to yz-plane.

Accepted Answer

A) -5 
B)  $\sqrt { 34 }$ 
C) -4 
D) -3 
E) 5
F) $5 \sqrt { 2 }$ 
G)4
H)3 
A) -5 
B)  $\sqrt { 34 }$ 
C) -4 
D) -3 
E) 5
F) $5 \sqrt { 2 }$ 
G)4
H)3

Question 11

Find parametric equations and symmetric equations of the line passing through the points A and B shown below.

Accepted Answer

x = t, y =

Question 12

Find the distance from the center of the sphere whose equation is $x ^ { 2 } + 2 x + y ^ { 2 } + 4 y + z ^ { 2 } + 6 z = 0$ to the xz-plane.

Accepted Answer

A)  $\sqrt { 14 }$ 
B) 4 
C)  $\sqrt { 10 }$ 
D) 3 
E) 1
F) $\sqrt { 13 }$ 
G)2
H) $\sqrt { 5 }$ 
A)  $\sqrt { 14 }$ 
B) 4 
C)  $\sqrt { 10 }$ 
D) 3 
E) 1
F) $\sqrt { 13 }$ 
G)2
H) $\sqrt { 5 }$

Question 13

Given three points P(1, -1, 0), Q(0, 1, 2) and R(-1, -1, 1), find the distance from the point P to the line passing through the points Q and R.

Accepted Answer

A)  $\sqrt { 29 }$ 
B)  $\sqrt { \frac { 29 } { 5 } }$ 
C)  $\sqrt { \frac { 29 } { 6 } }$ 
D)  $\frac { \sqrt { 29 } } { 3 }$ 
E)  $\frac { 1 } { \sqrt { 29 } }$ 
F) $\sqrt { \frac { 5 } { 29 } }$ 
G) $\sqrt { \frac { 6 } { 29 } }$ 
H) $\frac { 3 } { \sqrt { 29 } }$ 
A)  $\sqrt { 29 }$ 
B)  $\sqrt { \frac { 29 } { 5 } }$ 
C)  $\sqrt { \frac { 29 } { 6 } }$ 
D)  $\frac { \sqrt { 29 } } { 3 }$ 
E)  $\frac { 1 } { \sqrt { 29 } }$ 
F) $\sqrt { \frac { 5 } { 29 } }$ 
G) $\sqrt { \frac { 6 } { 29 } }$ 
H) $\frac { 3 } { \sqrt { 29 } }$

Question 14

Find the center of the sphere whose equation is $x ^ { 2 } + 2 x + y ^ { 2 } - y + z ^ { 2 } = 0$ .

Accepted Answer

A) (2, 1, 0) 
B)  $( 2 , - 1,0 )$ 
C)  $( - 2,1,0 )$ 
D)  $( - 2 , - 1,0 )$ 
E)  $\left( 1 , \frac { 1 } { 2 } , 0 \right)$ 
F) $\left( - 1 , \frac { 1 } { 2 } , 0 \right)$ 
G) $\left( 1 , - \frac { 1 } { 2 } , 0 \right)$ 
H) $\left( - 1 , - \frac { 1 } { 2 } , 0 \right)$ 
A) (2, 1, 0) 
B)  $( 2 , - 1,0 )$ 
C)  $( - 2,1,0 )$ 
D)  $( - 2 , - 1,0 )$ 
E)  $\left( 1 , \frac { 1 } { 2 } , 0 \right)$ 
F) $\left( - 1 , \frac { 1 } { 2 } , 0 \right)$ 
G) $\left( 1 , - \frac { 1 } { 2 } , 0 \right)$ 
H) $\left( - 1 , - \frac { 1 } { 2 } , 0 \right)$

Question 15

Find a unit vector that has the same direction as $( 3 , - 5 )$ .

Accepted Answer

The answer of Find a unit vector that has the...

Question 16

Describe the surface whose equation in cylindrical coordinates is r = 3.

Accepted Answer

A) Cylinder with vertical axis 
B) Cylinder with horizontal axis 
C) Sphere 
D) Vertical plane or half-plane 
E) Horizontal plane or half-plane
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) Vertical plane or half-plane 
E) Horizontal plane or half-plane
F)Paraboloid
G)Cone or half-cone with vertical axis
H)Cone or half-cone with horizontal axis

Question 17

Let f(x, y) = x sin y. Find f $\left( 2 , \frac { \pi } { 3 } \right)$ .

Accepted Answer

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

Question 18

Find the distance from the point (4, -3, 5) to the x-axis.

Accepted Answer

A) 5 
B)  $2 \sqrt { 5 }$ 
C)  $\sqrt { 34 }$ 
D)  $\sqrt { 5 }$ 
E) 50

F)3
G) $\sqrt { 41 }$ 
H)4 
A) 5 
B)  $2 \sqrt { 5 }$ 
C)  $\sqrt { 34 }$ 
D)  $\sqrt { 5 }$ 
E) 50

F)3
G) $\sqrt { 41 }$ 
H)4

Question 19

Given three points P(1, -1, 0), Q(0, 1, 2) and R(-1, -1, 1), find the distance from the point R to the line passing through the points Q and P.

Accepted Answer

A)  $\sqrt { 29 }$ 
B)  $\sqrt { \frac { 29 } { 5 } }$ 
C)  $\sqrt { \frac { 29 } { 6 } }$ 
D)  $\frac { \sqrt { 29 } } { 3 }$ 
E)  $\frac { 1 } { \sqrt { 29 } }$ 
F) $\sqrt { \frac { 5 } { 29 } }$ 
G) $\sqrt { \frac { 6 } { 29 } }$ 
H) $\frac { 3 } { \sqrt { 29 } }$ 
A)  $\sqrt { 29 }$ 
B)  $\sqrt { \frac { 29 } { 5 } }$ 
C)  $\sqrt { \frac { 29 } { 6 } }$ 
D)  $\frac { \sqrt { 29 } } { 3 }$ 
E)  $\frac { 1 } { \sqrt { 29 } }$ 
F) $\sqrt { \frac { 5 } { 29 } }$ 
G) $\sqrt { \frac { 6 } { 29 } }$ 
H) $\frac { 3 } { \sqrt { 29 } }$

Question 20

Given a = $\langle1,0,1 \rangle$ and b = $\langle 0,1 , - 1 \rangle$ , find the vector projection of a onto b.

Accepted Answer

A)  $\langle 0,1 , - 1 \rangle$ 
B)  $\langle 0 , - 1,1 \rangle$ 
C)  $\langle 0 , - \frac { 1 } { 2 } , \frac { 1 } { 2 } \rangle$ 
D)  $\langle0 , \frac { 1 } { 2 } , - \frac { 1 } { 2 } \rangle$ 
E)  $\langle 1,0,1 \rangle$ 
F) $\langle - 1,0 , - 1 \rangle$ 
G) $\left\langle - \frac { 1 } { 2 } , 0 , - \frac { 1 } { 2 } \right\rangle$ 
H) $\left\langle\frac { 1 } { 2 } , 0 , \frac { 1 } { 2 } \right\rangle$ 
A)  $\langle 0,1 , - 1 \rangle$ 
B)  $\langle 0 , - 1,1 \rangle$ 
C)  $\langle 0 , - \frac { 1 } { 2 } , \frac { 1 } { 2 } \rangle$ 
D)  $\langle0 , \frac { 1 } { 2 } , - \frac { 1 } { 2 } \rangle$ 
E)  $\langle 1,0,1 \rangle$ 
F) $\langle - 1,0 , - 1 \rangle$ 
G) $\left\langle - \frac { 1 } { 2 } , 0 , - \frac { 1 } { 2 } \right\rangle$ 
H) $\left\langle\frac { 1 } { 2 } , 0 , \frac { 1 } { 2 } \right\rangle$

Find an equation of the surface consisting of all points that are equidistant from the point (0, 1, 0) and the plane y = -1. Then sketch the surface.

What region of $\mathbb { R } ^ { 3 }$ is represented by the equation z = 3?

Convert $( 1 , \pi , 1 )$ from cylindrical coordinates to rectangular coordinates.

Find equations of all planes containing z-axis.

Find cylindrical and spherical equations for the surface whose equation in rectangular coordinates is x = 2. Describe the surface.

Find an equation of a plane containing the point P(2, -1, 1) and the y-axis.

Find rectangular and cylindrical equations for the surface whose equation in spherical coordinates is $\beta = 3$ . Describe the surface.

Find rectangular and spherical equations for the surface whose equation in cylindrical coordinates is $\theta = \frac { \pi } { 4 }$ . Describe the surface.

Describe the surface whose equation in spherical coordinates is $\sin \phi = \cos \phi$ .

Find the distance from the point (4, -3, 5) to yz-plane.

Find parametric equations and symmetric equations of the line passing through the points A and B shown below.

Find the distance from the center of the sphere whose equation is $x ^ { 2 } + 2 x + y ^ { 2 } + 4 y + z ^ { 2 } + 6 z = 0$ to the xz-plane.

Given three points P(1, -1, 0), Q(0, 1, 2) and R(-1, -1, 1), find the distance from the point P to the line passing through the points Q and R.

Find the center of the sphere whose equation is $x ^ { 2 } + 2 x + y ^ { 2 } - y + z ^ { 2 } = 0$ .

Find a unit vector that has the same direction as $( 3 , - 5 )$ .

Describe the surface whose equation in cylindrical coordinates is r = 3.

Let f(x, y) = x sin y. Find f $\left( 2 , \frac { \pi } { 3 } \right)$ .

Find the distance from the point (4, -3, 5) to the x-axis.

Given three points P(1, -1, 0), Q(0, 1, 2) and R(-1, -1, 1), find the distance from the point R to the line passing through the points Q and P.

Given a = $\langle1,0,1 \rangle$ and b = $\langle 0,1 , - 1 \rangle$ , find the vector projection of a onto b.

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Differential Equations

Infinite Sequences and Series

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Partial Derivatives

Multiple Integrals

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Exam 9: Vectors and the Geometry of Space

Find an equation of the surface consisting of all points that are equidistant from the point (0, 1, 0) and the plane y = -1. Then sketch the surface.

What region of R3\mathbb { R } ^ { 3 }R3 is represented by the equation z = 3?

Convert (1,π,1)( 1 , \pi , 1 )(1,π,1) from cylindrical coordinates to rectangular coordinates.

Find equations of all planes containing z-axis.

Find cylindrical and spherical equations for the surface whose equation in rectangular coordinates is x = 2. Describe the surface.

Find an equation of a plane containing the point P(2, -1, 1) and the y-axis.

Find rectangular and cylindrical equations for the surface whose equation in spherical coordinates is β=3\beta = 3β=3 . Describe the surface.

Find rectangular and spherical equations for the surface whose equation in cylindrical coordinates is θ=π4\theta = \frac { \pi } { 4 }θ=4π​ . Describe the surface.

Describe the surface whose equation in spherical coordinates is sin⁡ϕ=cos⁡ϕ\sin \phi = \cos \phisinϕ=cosϕ .

Find the distance from the point (4, -3, 5) to yz-plane.

Find parametric equations and symmetric equations of the line passing through the points A and B shown below.

Find the distance from the center of the sphere whose equation is x2+2x+y2+4y+z2+6z=0x ^ { 2 } + 2 x + y ^ { 2 } + 4 y + z ^ { 2 } + 6 z = 0x2+2x+y2+4y+z2+6z=0 to the xz-plane.

Given three points P(1, -1, 0), Q(0, 1, 2) and R(-1, -1, 1), find the distance from the point P to the line passing through the points Q and R.

Find the center of the sphere whose equation is x2+2x+y2−y+z2=0x ^ { 2 } + 2 x + y ^ { 2 } - y + z ^ { 2 } = 0x2+2x+y2−y+z2=0 .

Find a unit vector that has the same direction as (3,−5)( 3 , - 5 )(3,−5) .

Describe the surface whose equation in cylindrical coordinates is r = 3.

Let f(x, y) = x sin y. Find f (2,π3)\left( 2 , \frac { \pi } { 3 } \right)(2,3π​) .

Find the distance from the point (4, -3, 5) to the x-axis.

Given three points P(1, -1, 0), Q(0, 1, 2) and R(-1, -1, 1), find the distance from the point R to the line passing through the points Q and P.

Given a = ⟨1,0,1⟩\langle1,0,1 \rangle⟨1,0,1⟩ and b = ⟨0,1,−1⟩\langle 0,1 , - 1 \rangle⟨0,1,−1⟩ , find the vector projection of a onto b.

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

What region of $\mathbb { R } ^ { 3 }$ is represented by the equation z = 3?

Convert $( 1 , \pi , 1 )$ from cylindrical coordinates to rectangular coordinates.

Find rectangular and cylindrical equations for the surface whose equation in spherical coordinates is $\beta = 3$ . Describe the surface.

Find rectangular and spherical equations for the surface whose equation in cylindrical coordinates is $\theta = \frac { \pi } { 4 }$ . Describe the surface.

Describe the surface whose equation in spherical coordinates is $\sin \phi = \cos \phi$ .

Find the distance from the center of the sphere whose equation is $x ^ { 2 } + 2 x + y ^ { 2 } + 4 y + z ^ { 2 } + 6 z = 0$ to the xz-plane.

Find the center of the sphere whose equation is $x ^ { 2 } + 2 x + y ^ { 2 } - y + z ^ { 2 } = 0$ .

Find a unit vector that has the same direction as $( 3 , - 5 )$ .

Let f(x, y) = x sin y. Find f $\left( 2 , \frac { \pi } { 3 } \right)$ .

Given a = $\langle1,0,1 \rangle$ and b = $\langle 0,1 , - 1 \rangle$ , find the vector projection of a onto b.