Question 1

The symmetric equations of the line through (3,2,1) in the direction of $3 \mathbf { i } - \mathbf { j } + \mathbf { k }$ is

Accepted Answer

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

Question 2

In the three-dimensional Euclidean space, the graph of $x ^ { 2 } + \frac { y ^ { 2 } } { 16 } - z ^ { 2 } = - 1$ is

Accepted Answer

A) An elliptic cone 
B) A hyperboloid of one sheet 
C) A circular cylinder 
D) A hyperboloid of two sheets 
E) An elliptic paraboloid 
A) An elliptic cone 
B) A hyperboloid of one sheet 
C) A circular cylinder 
D) A hyperboloid of two sheets 
E) An elliptic paraboloid

Question 3

In the three-dimensional Euclidean space, the graph of $z = \frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 9 }$ is

Accepted Answer

A) An elliptic cylinder 
B) A hyperboloid of one sheet 
C) An elliptic paraboloid 
D) A hyperboloid of two sheets 
E) An ellipsoid 
A) An elliptic cylinder 
B) A hyperboloid of one sheet 
C) An elliptic paraboloid 
D) A hyperboloid of two sheets 
E) An ellipsoid

Question 4

The center and radius of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x = 0$ are

Accepted Answer

A)  $( - 2,0,0 ) ; 2$ 
B)  $( 2,0,0 ) ; 2$ 
C)  $( - 2,0,0 ) ; 1$ 
D)  $( 2,0,0 ) ; 4$ 
E)  $( - 2,0,0 ) ; 4$ 
A)  $( - 2,0,0 ) ; 2$ 
B)  $( 2,0,0 ) ; 2$ 
C)  $( - 2,0,0 ) ; 1$ 
D)  $( 2,0,0 ) ; 4$ 
E)  $( - 2,0,0 ) ; 4$

Question 5

The angle, in degrees and correct to two decimal places, between $\mathbf { v } = 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k }$ and $\mathbf { w } = \mathbf { i } - \mathbf { j } + \mathbf { k }$ is

Accepted Answer

A) 25.24 
B) 24.38 
C) 22.21 
D) 21.68 
E) 20.77 
A) 25.24 
B) 24.38 
C) 22.21 
D) 21.68 
E) 20.77

Question 6

The equation of the plane through (2,-1,3) parallel to the xy-plane is

Accepted Answer

A)  $z = 3$ 
B)  $x = 2$ 
C)  $y = - 1$ 
D)  $z = - 3$ 
E)  $y = 1$ 
A)  $z = 3$ 
B)  $x = 2$ 
C)  $y = - 1$ 
D)  $z = - 3$ 
E)  $y = 1$

Question 7

The symmetric equations of the line through (1,2,-1) perpendicular to each of the vectors $2 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k }$ and $- 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }$ is

Accepted Answer

A)  $x = 1 , \frac { y + 2 } { 2 } = \frac { z + 1 } { 4 }$ 
B)  $x = 1 , \frac { y - 2 } { 2 } = \frac { z - 1 } { 4 }$ 
C)  $x = 1 , \frac { y - 2 } { 2 } = \frac { z + 1 } { 4 }$ 
D)  $x = 1 , \frac { y + 2 } { 2 } = \frac { z - 1 } { 4 }$ 
E)  $x = 1 , \frac { y - 2 } { - 2 } = \frac { z + 1 } { 4 }$ 
A)  $x = 1 , \frac { y + 2 } { 2 } = \frac { z + 1 } { 4 }$ 
B)  $x = 1 , \frac { y - 2 } { 2 } = \frac { z - 1 } { 4 }$ 
C)  $x = 1 , \frac { y - 2 } { 2 } = \frac { z + 1 } { 4 }$ 
D)  $x = 1 , \frac { y + 2 } { 2 } = \frac { z - 1 } { 4 }$ 
E)  $x = 1 , \frac { y - 2 } { - 2 } = \frac { z + 1 } { 4 }$

Question 8

The parametric equations of the line of intersection of the planes $x - 2 y + 4 z = 2$ and $x + y - 2 z = 5$ are

Accepted Answer

A)  $x = 4 , y = 2 t + 1 , z = - t$ 
B)  $x = 2 , y = - 2 t + 1 , z = t + 1$ 
C)  $x = 4 , y = 2 t + 1 , z = t - 1$ 
D)  $x = 4 , y = 2 t + 1 , z = t$ 
E)  $x = 2 , y = 2 t - 3 , z = t + 1$ 
A)  $x = 4 , y = 2 t + 1 , z = - t$ 
B)  $x = 2 , y = - 2 t + 1 , z = t + 1$ 
C)  $x = 4 , y = 2 t + 1 , z = t - 1$ 
D)  $x = 4 , y = 2 t + 1 , z = t$ 
E)  $x = 2 , y = 2 t - 3 , z = t + 1$

Question 9

The point of intersection of the line $x = 2 t + 1 , y = 5 t + 1 , z = 3 t$ and the plane $x + y + z = 2$ is

Accepted Answer

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

Question 10

The distance between $P _ { 1 } = ( 0,4,1 )$ and $P _ { 2 } = ( 1,8,2 )$ is

Accepted Answer

A)  $\sqrt { 17 }$ 
B)  $3 \sqrt { 2 }$ 
C)  $\sqrt { 19 }$ 
D)  $\sqrt { 21 }$ 
E)  $\sqrt { 23 }$ 
A)  $\sqrt { 17 }$ 
B)  $3 \sqrt { 2 }$ 
C)  $\sqrt { 19 }$ 
D)  $\sqrt { 21 }$ 
E)  $\sqrt { 23 }$

Question 11

In the three-dimensional Euclidean space, the graph of $4 x ^ { 2 } + y ^ { 2 } = 16$ is

Accepted Answer

A) An elliptic cylinder 
B) A sphere 
C) A parabolic cylinder 
D) A circular cylinder 
E) An ellipsoid 
A) An elliptic cylinder 
B) A sphere 
C) A parabolic cylinder 
D) A circular cylinder 
E) An ellipsoid

Question 12

The value of a that makes $\mathbf { v } = \mathbf { i } + \mathbf { j } - \mathbf { k }$ and $\mathbf { w } = 5 \mathbf { i } + a \mathbf { j } + \mathbf { k }$ orthogonal is

Accepted Answer

A) 8 
B) 6 
C) 2 
D) -2 
E) -8 
A) 8 
B) 6 
C) 2 
D) -2 
E) -8

Question 13

In the three-dimensional Euclidean space, the graph of $z = \frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 }$ is

Accepted Answer

A) An elliptic cone 
B) A hyperboloid of one sheet 
C) A hyperbolic paraboloid 
D) A hyperboloid of two sheets 
E) An elliptic paraboloid 
A) An elliptic cone 
B) A hyperboloid of one sheet 
C) A hyperbolic paraboloid 
D) A hyperboloid of two sheets 
E) An elliptic paraboloid

Question 14

Let , and $\mathbf { u } = \langle 1,1 , - 6 \rangle$ Then $\mathbf { v } = \langle 3,0,8 \rangle$ is

Accepted Answer

A)  $\sqrt { 119 } - \sqrt { 14 }$ 
B)  $\sqrt { 203 } - \sqrt { 14 }$ 
C)  $\sqrt { 201 } - \sqrt { 14 }$ 
D)  $\sqrt { 201 } - \sqrt { 13 }$ 
E)  $\sqrt { 203 } - \sqrt { 13 }$ 
A)  $\sqrt { 119 } - \sqrt { 14 }$ 
B)  $\sqrt { 203 } - \sqrt { 14 }$ 
C)  $\sqrt { 201 } - \sqrt { 14 }$ 
D)  $\sqrt { 201 } - \sqrt { 13 }$ 
E)  $\sqrt { 203 } - \sqrt { 13 }$

Question 15

The equation of the plane through (2,3,-1) normal to the line $\frac { x - 1 } { 2 } = \frac { y - 1 } { 3 } = \frac { z + 1 } { - 2 }$ is

Accepted Answer

A)  $2 x + 3 y - 2 z = 14$ 
B)  $2 x + 3 y - 2 z = 15$ 
C)  $2 x + 3 y - 2 z = - 14$ 
D)  $2 x + 3 y - 2 z = - 15$ 
E)  $2 x + 3 y - 2 z = 13$ 
A)  $2 x + 3 y - 2 z = 14$ 
B)  $2 x + 3 y - 2 z = 15$ 
C)  $2 x + 3 y - 2 z = - 14$ 
D)  $2 x + 3 y - 2 z = - 15$ 
E)  $2 x + 3 y - 2 z = 13$

Question 16

The equation of the plane through (-3,1,4) parallel to the yz-plane is

Accepted Answer

A)  $z = 4$ 
B)  $y = 1$ 
C)  $y = x + 4$ 
D)  $x = - 3$ 
E)  $z = x + 7$ 
A)  $z = 4$ 
B)  $y = 1$ 
C)  $y = x + 4$ 
D)  $x = - 3$ 
E)  $z = x + 7$

Question 17

Let , and $\langle \mathbf { u } = 1,1 , - 6 , \rangle$ Then $\mathbf { v } = \langle 3,0,8 \rangle$ is

Accepted Answer

A)  $\langle - 25,2 , - 44 \rangle$ 
B)  $\langle - 25,2,44 \rangle$ 
C)  $\langle 25 , - 2 , - 44 \rangle$ 
D)  $\langle - 25 , - 2 , - 44 \rangle$ 
E)  $\langle - 25 , - 2,44 \rangle$ 
A)  $\langle - 25,2 , - 44 \rangle$ 
B)  $\langle - 25,2,44 \rangle$ 
C)  $\langle 25 , - 2 , - 44 \rangle$ 
D)  $\langle - 25 , - 2 , - 44 \rangle$ 
E)  $\langle - 25 , - 2,44 \rangle$

Question 18

The distance between $P _ { 1 } = ( 5,6 , - 1 )$ and $P _ { 2 } = ( 3,8,2 )$ is

Accepted Answer

A)  $\sqrt { 15 }$ 
B) 4 
C)  $\sqrt { 17 }$ 
D)  $3 \sqrt { 2 }$ 
E)  $\sqrt { 19 }$ 
A)  $\sqrt { 15 }$ 
B) 4 
C)  $\sqrt { 17 }$ 
D)  $3 \sqrt { 2 }$ 
E)  $\sqrt { 19 }$

Question 19

Let $P _ { 1 } = ( 3 , - 2,5 )$ and $P _ { 2 } = ( 10,4,3 )$ If v is $\overrightarrow { P _ { 1 } P _ { 2 } }$ in standard position, then v is

Accepted Answer

A)  $\langle 7 , - 6 , - 2 \rangle$ 
B)  $\langle 7 , - 6,2 \rangle$ 
C)  $\langle 7,6 , - 2 \rangle$ 
D)  $\langle 7,6,2 \rangle$ 
E)  $\langle - 7,6 , - 2 \rangle$ 
A)  $\langle 7 , - 6 , - 2 \rangle$ 
B)  $\langle 7 , - 6,2 \rangle$ 
C)  $\langle 7,6 , - 2 \rangle$ 
D)  $\langle 7,6,2 \rangle$ 
E)  $\langle - 7,6 , - 2 \rangle$

Question 20

The distance from the point $( 2 , - 3,4 )$ to the plane $x + 2 y + 2 z = 13$ is

Accepted Answer

A) 2 
B) 3 
C) 5 
D) 6 
E) 7 
A) 2 
B) 3 
C) 5 
D) 6 
E) 7

The symmetric equations of the line through (3,2,1) in the direction of $3 \mathbf { i } - \mathbf { j } + \mathbf { k }$ is

In the three-dimensional Euclidean space, the graph of $x ^ { 2 } + \frac { y ^ { 2 } } { 16 } - z ^ { 2 } = - 1$ is

In the three-dimensional Euclidean space, the graph of $z = \frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 9 }$ is

The center and radius of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x = 0$ are

The angle, in degrees and correct to two decimal places, between $\mathbf { v } = 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k }$ and $\mathbf { w } = \mathbf { i } - \mathbf { j } + \mathbf { k }$ is

The equation of the plane through (2,-1,3) parallel to the xy-plane is

The symmetric equations of the line through (1,2,-1) perpendicular to each of the vectors $2 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k }$ and $- 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }$ is

The parametric equations of the line of intersection of the planes $x - 2 y + 4 z = 2$ and $x + y - 2 z = 5$ are

The point of intersection of the line $x = 2 t + 1 , y = 5 t + 1 , z = 3 t$ and the plane $x + y + z = 2$ is

The distance between $P _ { 1 } = ( 0,4,1 )$ and $P _ { 2 } = ( 1,8,2 )$ is

In the three-dimensional Euclidean space, the graph of $4 x ^ { 2 } + y ^ { 2 } = 16$ is

The value of a that makes $\mathbf { v } = \mathbf { i } + \mathbf { j } - \mathbf { k }$ and $\mathbf { w } = 5 \mathbf { i } + a \mathbf { j } + \mathbf { k }$ orthogonal is

In the three-dimensional Euclidean space, the graph of $z = \frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 }$ is

Let , and $\mathbf { u } = \langle 1,1 , - 6 \rangle$ Then $\mathbf { v } = \langle 3,0,8 \rangle$ is

The equation of the plane through (2,3,-1) normal to the line $\frac { x - 1 } { 2 } = \frac { y - 1 } { 3 } = \frac { z + 1 } { - 2 }$ is

The equation of the plane through (-3,1,4) parallel to the yz-plane is

Let , and $\langle \mathbf { u } = 1,1 , - 6 , \rangle$ Then $\mathbf { v } = \langle 3,0,8 \rangle$ is

The distance between $P _ { 1 } = ( 5,6 , - 1 )$ and $P _ { 2 } = ( 3,8,2 )$ is

Let $P _ { 1 } = ( 3 , - 2,5 )$ and $P _ { 2 } = ( 10,4,3 )$ If v is $\overrightarrow { P _ { 1 } P _ { 2 } }$ in standard position, then v is

The distance from the point $( 2 , - 3,4 )$ to the plane $x + 2 y + 2 z = 13$ is

Preparing for Calculus

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Exam 11: Vectors; Lines, Planes, and Quadric Surfaces in Space

The symmetric equations of the line through (3,2,1) in the direction of 3i−j+k3 \mathbf { i } - \mathbf { j } + \mathbf { k }3i−j+k is

In the three-dimensional Euclidean space, the graph of x2+y216−z2=−1x ^ { 2 } + \frac { y ^ { 2 } } { 16 } - z ^ { 2 } = - 1x2+16y2​−z2=−1 is

In the three-dimensional Euclidean space, the graph of z=x24+y29z = \frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 9 }z=4x2​+9y2​ is

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

The angle, in degrees and correct to two decimal places, between v=2i−3j+k\mathbf { v } = 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k }v=2i−3j+k and w=i−j+k\mathbf { w } = \mathbf { i } - \mathbf { j } + \mathbf { k }w=i−j+k is

The equation of the plane through (2,-1,3) parallel to the xy-plane is

The symmetric equations of the line through (1,2,-1) perpendicular to each of the vectors 2i+4j−2k2 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k }2i+4j−2k and −3i−2j+k- 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }−3i−2j+k is

The parametric equations of the line of intersection of the planes x−2y+4z=2x - 2 y + 4 z = 2x−2y+4z=2 and x+y−2z=5x + y - 2 z = 5x+y−2z=5 are

The point of intersection of the line x=2t+1,y=5t+1,z=3tx = 2 t + 1 , y = 5 t + 1 , z = 3 tx=2t+1,y=5t+1,z=3t and the plane x+y+z=2x + y + z = 2x+y+z=2 is

The distance between P1=(0,4,1)P _ { 1 } = ( 0,4,1 )P1​=(0,4,1) and P2=(1,8,2)P _ { 2 } = ( 1,8,2 )P2​=(1,8,2) is

In the three-dimensional Euclidean space, the graph of 4x2+y2=164 x ^ { 2 } + y ^ { 2 } = 164x2+y2=16 is

The value of a that makes v=i+j−k\mathbf { v } = \mathbf { i } + \mathbf { j } - \mathbf { k }v=i+j−k and w=5i+aj+k\mathbf { w } = 5 \mathbf { i } + a \mathbf { j } + \mathbf { k }w=5i+aj+k orthogonal is

In the three-dimensional Euclidean space, the graph of z=x216−y29z = \frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 }z=16x2​−9y2​ is

Let , and u=⟨1,1,−6⟩\mathbf { u } = \langle 1,1 , - 6 \rangleu=⟨1,1,−6⟩ Then v=⟨3,0,8⟩\mathbf { v } = \langle 3,0,8 \ranglev=⟨3,0,8⟩ is

The equation of the plane through (2,3,-1) normal to the line x−12=y−13=z+1−2\frac { x - 1 } { 2 } = \frac { y - 1 } { 3 } = \frac { z + 1 } { - 2 }2x−1​=3y−1​=−2z+1​ is

The equation of the plane through (-3,1,4) parallel to the yz-plane is

Let , and ⟨u=1,1,−6,⟩\langle \mathbf { u } = 1,1 , - 6 , \rangle⟨u=1,1,−6,⟩ Then v=⟨3,0,8⟩\mathbf { v } = \langle 3,0,8 \ranglev=⟨3,0,8⟩ is

The distance between P1=(5,6,−1)P _ { 1 } = ( 5,6 , - 1 )P1​=(5,6,−1) and P2=(3,8,2)P _ { 2 } = ( 3,8,2 )P2​=(3,8,2) is

Let P1=(3,−2,5)P _ { 1 } = ( 3 , - 2,5 )P1​=(3,−2,5) and P2=(10,4,3)P _ { 2 } = ( 10,4,3 )P2​=(10,4,3) If v is P1P2→\overrightarrow { P _ { 1 } P _ { 2 } }P1​P2​​ in standard position, then v is

The distance from the point (2,−3,4)( 2 , - 3,4 )(2,−3,4) to the plane x+2y+2z=13x + 2 y + 2 z = 13x+2y+2z=13 is

Preparing for Calculus

Limits and Continuity

The Derivative

More About Derivatives

Applications of the Derivative

The Integral

Applications of the Integral

Techniques of Integration

Infinite Series

Parametric Equations; Polar Equations

Vector Functions

Functions of Several Variables

Directional Derivatives, Gradients, and Extrema

Multiple Integrals

Vector Calculus

Differential Equations

Filters

The symmetric equations of the line through (3,2,1) in the direction of $3 \mathbf { i } - \mathbf { j } + \mathbf { k }$ is

In the three-dimensional Euclidean space, the graph of $x ^ { 2 } + \frac { y ^ { 2 } } { 16 } - z ^ { 2 } = - 1$ is

In the three-dimensional Euclidean space, the graph of $z = \frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 9 }$ is

The center and radius of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x = 0$ are

The angle, in degrees and correct to two decimal places, between $\mathbf { v } = 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k }$ and $\mathbf { w } = \mathbf { i } - \mathbf { j } + \mathbf { k }$ is

The symmetric equations of the line through (1,2,-1) perpendicular to each of the vectors $2 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k }$ and $- 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }$ is

The parametric equations of the line of intersection of the planes $x - 2 y + 4 z = 2$ and $x + y - 2 z = 5$ are

The point of intersection of the line $x = 2 t + 1 , y = 5 t + 1 , z = 3 t$ and the plane $x + y + z = 2$ is

The distance between $P _ { 1 } = ( 0,4,1 )$ and $P _ { 2 } = ( 1,8,2 )$ is

In the three-dimensional Euclidean space, the graph of $4 x ^ { 2 } + y ^ { 2 } = 16$ is

The value of a that makes $\mathbf { v } = \mathbf { i } + \mathbf { j } - \mathbf { k }$ and $\mathbf { w } = 5 \mathbf { i } + a \mathbf { j } + \mathbf { k }$ orthogonal is

In the three-dimensional Euclidean space, the graph of $z = \frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 }$ is

Let , and $\mathbf { u } = \langle 1,1 , - 6 \rangle$ Then $\mathbf { v } = \langle 3,0,8 \rangle$ is

The equation of the plane through (2,3,-1) normal to the line $\frac { x - 1 } { 2 } = \frac { y - 1 } { 3 } = \frac { z + 1 } { - 2 }$ is

Let , and $\langle \mathbf { u } = 1,1 , - 6 , \rangle$ Then $\mathbf { v } = \langle 3,0,8 \rangle$ is

The distance between $P _ { 1 } = ( 5,6 , - 1 )$ and $P _ { 2 } = ( 3,8,2 )$ is

Let $P _ { 1 } = ( 3 , - 2,5 )$ and $P _ { 2 } = ( 10,4,3 )$ If v is $\overrightarrow { P _ { 1 } P _ { 2 } }$ in standard position, then v is

The distance from the point $( 2 , - 3,4 )$ to the plane $x + 2 y + 2 z = 13$ is