Deck 9: Vectors and the Geometry of Space

Question 1

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

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

Answer 1

A

Question 2

Describe the surface whose equation in cylindrical coordinates is $z = r ^ { 2 }$ .

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

Answer 2

Paraboloid

Question 3

If in cylindrical coordinates, find rectangular coordinates of Q.

Answer 3

(0, 1, 3)

Question 4

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

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

Question 5

Describe the surface whose equation in cylindrical coordinates is $z = 4 r$ .

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 6

Convert $( 1,1 , \sqrt { 2 } )$ from rectangular coordinates to spherical coordinates.

A) $\left( \sqrt { 2 } , \frac { \pi } { 4 } , \frac { \pi } { 4 } \right)$
B) $\left( 2 , \frac { \pi } { 4 } , \frac { \pi } { 4 } \right)$
C) $\left( \sqrt { 2 } , \frac { \pi } { 2 } , \frac { \pi } { 4 } \right)$
D) $\left( 2 , \frac { \pi } { 2 } , \frac { \pi } { 4 } \right)$
E) $\left( \sqrt { 2 } , \frac { \pi } { 4 } , \frac { \pi } { 2 } \right)$
F) $\left( 2 , \frac { \pi } { 4 } , \frac { \pi } { 2 } \right)$
G) $\left( \sqrt { 2 } , \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$
H) $\left( 2 , \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$

Question 7

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

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 8

Describe the surface whose equation in cylindrical coordinates is $\rho = 3 \sec \phi$ .

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 9

Convert $\left( 1 , \frac { \pi } { 4 } , \frac { \pi } { 4 } \right)$ from spherical coordinates to rectangular coordinates.

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

Question 10

Describe the surface whose equation in cylindrical coordinates is $\rho = 4 \cos \phi$ .

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 11

Convert $( - 1 , \sqrt { 3 } , 2 )$ from rectangular coordinates to spherical coordinates.

A) $\left( 2 , \frac { \pi } { 6 } , \frac { \pi } { 4 } \right)$
B) $\left( 4 , \frac { \pi } { 6 } , \frac { \pi } { 4 } \right)$
C) $\left( \sqrt { 2 } , \frac { \pi } { 6 } , \frac { \pi } { 4 } \right)$
D) $\left( \sqrt { 8 } , \frac { 2 \pi } { 3 } , \frac { \pi } { 4 } \right)$
E) $\left( 2 , \frac { \pi } { 3 } , \frac { \pi } { 4 } \right)$
F) $\left( 4 , \frac { \pi } { 3 } , \frac { \pi } { 4 } \right)$
G) $\left( \sqrt { 2 } , \frac { 4 \pi } { 3 } , \frac { \pi } { 4 } \right)$
H) $\left( \sqrt { 8 } , \frac { \pi } { 3 } , \frac { \pi } { 4 } \right)$

Question 12

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

A) $\left( \sqrt { 2 } , \frac { \pi } { 2 } , 1 \right)$
B) $\left( \sqrt { 2 } , \frac { \pi } { 4 } , 1 \right)$
C) $\left( 1 , \frac { \pi } { 2 } , 1 \right)$
D) $\left( 1 , \frac { \pi } { 4 } , 1 \right)$
E) $\left( 1 , \frac { \pi } { 2 } , \sqrt { 2 } \right)$
F) $\left( 1 , \frac { \pi } { 4 } , \sqrt { 2 } \right)$
G) $\left( 1 , \frac { \pi } { 2 } , 2 \right)$
H) $\left( 1 , \frac { \pi } { 4 } , 2 \right)$

Question 13

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

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 14

Convert $( 1 , - \sqrt { 3 } , \sqrt { 3 } )$ from rectangular coordinates to cylindrical coordinates.

A) $\left( 1 , \frac { \pi } { 3 } , \sqrt { 3 } \right)$
B) $\left( 1 , \frac { \pi } { 6 } , \sqrt { 3 } \right)$
C) $\left( \sqrt { 3 } , \frac { \pi } { 3 } , 1 \right)$
D) $\left( \sqrt { 3 } , \frac { \pi } { 6 } , 1 \right)$
E) $\left( 2 , \frac { \pi } { 3 } , \sqrt { 3 } \right)$
F) $\left( 2 , - \frac { \pi } { 3 } , \sqrt { 3 } \right)$
G) $\left( \sqrt { 3 } , \frac { \pi } { 3 } , 2 \right)$
H) $\left( \sqrt { 3 } , \frac { \pi } { 6 } , 2 \right)$

Question 15

If in rectangular coordinates, find the spherical coordinates of P.

Question 16

Describe the surface whose equation in cylindrical coordinates is $\phi = 3$ .

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

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

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

Question 18

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

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 19

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

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

Question 20

Describe the surface whose equation in cylindrical coordinates is $\beta = 3$ .

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 21

Convert the point (0, -5, 0) to cylindrical and spherical coordinates.

Question 22

Find the set of intersection of the surfaces whose equations in spherical coordinates are and .

Question 23

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

Question 24

Find the set of intersection of the surfaces whose equations in spherical coordinates are and .

Question 25

Describe the surface whose equation in cylindrical coordinates is .

Question 26

Describe the surface whose equation in spherical coordinates is .

Question 27

Let A, B, and C be three points on the unit sphere centered at the origin whose spherical coordinates are A = (1, 0, 0), B = , and C = .(a) Find the angle between and .(b) Find the angle between and .(c) Find the angle between and .

Question 28

Describe the surface whose equation in cylindrical coordinates is .

Question 29

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

Question 30

Find the set of intersection of the surfaces whose equations in spherical coordinates are and .

Question 31

Sketch the solid given in cylindrical coordinates by .

Question 32

Sketch the solid given in spherical coordinates by .

Question 33

Find the set of intersection of the surfaces whose equations in spherical coordinates are and .

Question 34

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

Question 35

Find the set of intersection of the surfaces whose equations in spherical coordinates are and .

Question 36

New Orleans is situated at latitude 30° N and longitude 90° W, and New York is situated at latitude 41° N and longitude 74° W. Find the distance from New Orleans to New York, assuming that the radius of the earth is 3960 miles.

Question 37

Describe in words or sketch the solid represented in cylindrical coordinates by the inequalities .

Question 38

Use the given data:
Los Angeles: Latitude 34.05°N and Longitude 118.25°W;
Hawaii: Latitude 21.3°N and Longitude 157.83°W.Find the distance from Los Angeles to Hawaii (Assume the radius of earth is 3960 miles.)

Question 39

Convert the point to cylindrical and spherical coordinates.

Question 40

Describe in words the solid represented in spherical coordinates by the inequality .

Question 41

Identify the trace of the surface $x ^ { 2 } = y ^ { 2 } + z ^ { 2 }$ in the plane z = 1.

A)ellipse but not a circle
B)parabola
C)hyperbola
D)circle
E)two parallel straight lines
F)two intersecting straight lines
G)point
H)straight line

Question 42

Identify the trace of the surface $x ^ { 2 } = y ^ { 2 } + z ^ { 2 }$ in the plane z = 0.

A)ellipse but not a circle
B)parabola
C)hyperbola
D)circle
E)two parallel straight lines
F)two intersecting straight lines
G)point
H)straight line

Question 43

Identify the trace of the surface $x = y ^ { 2 } + z ^ { 2 }$ in the plane x = y.

A)ellipse but not a circle
B)parabola
C)hyperbola
D)circle
E)two parallel straight lines
F)two intersecting straight lines
G)point
H)straight line

Question 44

Given points A = and B = in spherical coordinates, find the distance between the two points.

Question 45

Given points A = and B = in cylindrical coordinates, find the distance between the two points.

Question 46

Identify the trace of the surface $x = 2 y ^ { 2 } + 3 z ^ { 2 }$ in the plane x = 1.

A)ellipse but not a circle
B)parabola
C)hyperbola
D)circle
E)two parallel straight lines
F)two intersecting straight lines
G)point
H)straight line

Question 47

Identify the surface $x = y ^ { 2 } - z ^ { 2 }$ .

A)ellipsoid but not a sphere
B)hyperboloid of one sheet
C)hyperboloid of two sheets
D)cylinder
E)sphere
F)elliptic but not circular paraboloid
G)cone
H)hyperbolic paraboloid

Question 48

Identify the surface $x = y ^ { 2 } + 2 z ^ { 2 }$ .

A)ellipsoid but not a sphere
B)hyperboloid of one sheet
C)hyperboloid of two sheets
D)cylinder
E)sphere
F)elliptic but not circular paraboloid
G)cone
H)circular paraboloid (figure of revolution)

Question 49

Identify the surface $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 3$ .

A)ellipsoid but not a sphere
B)hyperboloid of one sheet
C)hyperboloid of two sheets
D)cylinder
E)sphere
F)elliptic but not circular paraboloid
G)cone
H)circular paraboloid (figure of revolution)

Question 50

Identify the surface $x ^ { 2 } - y ^ { 2 } + z ^ { 2 } = 10$ .

A)ellipsoid but not a sphere
B)hyperboloid of one sheet
C)hyperboloid of two sheets
D)cylinder
E)sphere
F)elliptic but not circular paraboloid
G)cone
H)circular paraboloid (figure of revolution)

Question 51

Identify the trace of the surface $x = y ^ { 2 } + z ^ { 2 }$ in the plane z = 1.

A)ellipse but not a circle
B)parabola
C)hyperbola
D)circle
E)two parallel straight lines
F)two intersecting straight lines
G)point
H)straight line

Question 52

Identify the trace of the surface $x ^ { 2 } = y ^ { 2 } + z ^ { 2 }$ in the plane x = 0.

A)ellipse but not a circle
B)parabola
C)hyperbola
D)circle
E)two parallel straight lines
F)two intersecting straight lines
G)point
H)straight line

Question 53

Identify the trace of the surface $x = y ^ { 2 } + z ^ { 2 }$ in the plane x = 0.

A)ellipse but not a circle
B)parabola
C)hyperbola
D)circle
E)two parallel straight lines
F)two intersecting straight lines
G)point
H)straight line

Question 54

Identify the trace of the surface $x ^ { 2 } = y ^ { 2 } + z ^ { 2 }$ in the plane y = 1.

A)ellipse but not a circle
B)parabola
C)hyperbola
D)circle
E)two parallel straight lines
F)two intersecting straight lines
G)point
H)straight line

Question 55

Identify the trace of the surface $x = y ^ { 2 } + z ^ { 2 }$ in the plane x = 1.

A)ellipse but not a circle
B)parabola
C)hyperbola
D)circle
E)two parallel straight lines
F)two intersecting straight lines
G)point
H)straight line

Question 56

Identify the surface $- x ^ { 2 } + y ^ { 2 } - z ^ { 2 } = 10$ .

A)ellipsoid but not a sphere
B)hyperboloid of one sheet
C)hyperboloid of two sheets
D)cylinder
E)sphere
F)elliptic but not circular paraboloid
G)cone
H)circular paraboloid (figure of revolution)

Question 57

Identify the surface $2 = y ^ { 2 } + z ^ { 2 }$ .

A)ellipsoid but not a sphere
B)hyperboloid of one sheet
C)hyperboloid of two sheets
D)cylinder
E)sphere
F)elliptic but not circular paraboloid
G)cone
H)circular paraboloid (figure of revolution)

Question 58

Identify the surface $x ^ { 2 } = y ^ { 2 } + z ^ { 2 }$ .

A)ellipsoid but not a sphere
B)hyperboloid of one sheet
C)hyperboloid of two sheets
D)cylinder
E)sphere
F)elliptic but not circular paraboloid
G)cone
H)circular paraboloid (figure of revolution)

Question 59

Identify the surface $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ .

A)ellipsoid but not a sphere
B)hyperboloid of one sheet
C)hyperboloid of two sheets
D)cylinder
E)sphere
F)elliptic but not circular paraboloid
G)cone
H)circular paraboloid (figure of revolution)

Question 60

Identify the trace of the surface $x = y ^ { 2 } + z ^ { 2 }$ in the plane y = 0.

A)ellipse but not a circle
B)parabola
C)hyperbola
D)circle
E)two parallel straight lines
F)two intersecting straight lines
G)point
H)straight line

Question 61

Let f(x, y) = $\left( x ^ { 2 } + y \right) ^ { 3 }$ . If x = 1, find f(x, 2x).

A)1
B)2
C)3
D)4
E)8
F)9
G)16
H)27

Question 62

Sketch and identify the quadric surface given by .

Question 63

Find the range of the function f(x, y) = $\sqrt { x - y ^ { 2 } }$ .

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

Question 64

Find the domain of the function f(x, y) = $e ^ { x - y ^ { 2 } }$ .

A)All points on or to the left of $x = y ^ { 2 }$
B)All points on or to the right of $x = y ^ { 2 }$
C)All points to the left of $x = y ^ { 2 }$
D)All points to the right of $x = y ^ { 2 }$
E)All points on or to the left of x = 0
F)All points on or to the right of x = 0
G)All points to the left of x = 0
H)All points in the xy-plane

Question 65

Describe the trace of the surface z = = 0 in the plane z = 1.

Question 66

Find the coordinates of the point(s) of intersection of the line x = 1 - t, y = 1 - t, z = 4t and the surface z = .

Question 67

Let f(x, y) = $x ^ { 2 } + 2 x y + y ^ { 2 }$ . If x = 2, find f(x, 2x).

A)12
B)16
C)24
D)28
E)32
F)36
G)42
H)48

Question 68

Let S be the quadric surface given by . What kind of surface is S?

Question 69

Let f(x, y) = x sin y. If x = $\pi$ , find f(x, x/2).

A) $\frac { \pi } { 6 }$
B) $\frac { \pi } { 4 }$
C) $\frac { \pi } { 3 }$
D) $\frac { \pi } { 2 }$
E) $\frac { 2 \pi } { 3 }$
F) $\frac { 3 \pi } { 4 }$
G) $\pi$
H) $2 \pi$

Question 70

Find the domain of the function f(x, y) = $\ln \left( x - y ^ { 2 } \right)$ .

A)All points on or to the left of $x = y ^ { 2 }$ e.All points on or to the left of x = 0
B)All points on or to the right of $x = y ^ { 2 }$ f.All points on or to the right of x = 0
C)All points to the left of $x = y ^ { 2 }$ g.All points to the left of x = 0
D)All points to the right of $x = y ^ { 2 }$ h.All points in the xy-plane

Question 71

Find the range of the function f(x, y) = $e ^ { x - y ^ { 2 } }$ .

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

Question 72

Find the range of the function f(x, y) = $\ln \left( x - y ^ { 2 } \right)$ .

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

Question 73

Let S be the quadric surface given by . What are the traces of S in each of the three coordinate planes?

Question 74

Which of the following is not a quadric surface?

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 75

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

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 76

Identify the graph of the function f(x, y) = $3 - x ^ { 2 } - y ^ { 2 }$ .

A)Cone
B)Paraboloid
C)Ellipsoid
D)Hyperboloid of one sheet
E)Hyperboloid of two sheets
F)Hyperbolic cylinder
G)Elliptic cylinder
H)Parabolic cylinder

Question 77

Sketch the graph of in , and name the surface.

Question 78

Find the domain of the function f(x, y) = $\sqrt { x - y ^ { 2 } }$ .

A)All points on or to the left of $x = y ^ { 2 }$ e.All points on or to the left of x = 0
B)All points on or to the right of $x = y ^ { 2 }$ f.All points on or to the right of x = 0
C)All points to the left of $x = y ^ { 2 }$ g.All points to the left of x = 0
D)All points to the right of $x = y ^ { 2 }$ h.All points in the xy-plane

Question 79

Sketch the graph of in , and name the surface.

Question 80

Describe the vertical traces x = 0 and the horizontal traces z = -1 (if any) for the surfaces and .