Deck 11: Three-Dimensional Space; Vectors

Question 1

Convert (4, , ) from spherical to rectangular coordinates.

A) (4, 4, 4)
B) (0, 4, 0)
C) (0, 0, 4)
D) (4, 0, 0)
E) (0, 0, 0)

Answer 1

(0, 4, 0)

Question 2

Convert (2, , ) from spherical to rectangular coordinates.

A) (2, 2, 2)
B) (0, -2, 0)
C) (0, 0, 2)
D) (2, 0, 0)
E) (0, 0, 0)

Answer 2

(0, -2, 0)

Question 3

Convert from rectangular to spherical coordinates.

A)
B)
C)
D)
E)

Answer 3

Question 4

Convert from cylindrical to rectangular coordinates.

A)
B)
C)
D)
E)

Question 5

Convert from rectangular to spherical coordinates.

A)
B)
C)
D)
E)

Question 6

Convert from spherical to rectangular coordinates.

A)
B)
C)
D)
E)

Question 7

Convert from spherical to rectangular coordinates.

A)
B)
C)
D)
E)

Question 8

The equation of a surface is given in rectangular coordinates. Give the equation of the surface in spherical coordinates.

Question 9

Convert from rectangular to spherical coordinates. Coordinates are written as .

A)
B)
C)
D)
E)

Question 10

Convert from spherical to rectangular coordinates.

A)
B)
C)
D)
E)

Question 11

Convert from rectangular to cylindrical coordinates.

A)
B)
C)
D)
E)

Question 12

Convert from rectangular to cylindrical coordinates.

A)
B)
C)
D)
E)

Question 13

The equation of a surface is given in rectangular coordinates. Give the equation of the surface in cylindrical coordinates.

Question 14

Convert from spherical to cylindrical coordinates.

A)
B)
C)
D)
E)

Question 15

Convert (1, 1, 10) from rectangular to cylindrical coordinates. Coordinates are written as .

A)
B)
C)
D)
E)

Question 16

Convert from cylindrical to spherical coordinates.

A)
B)
C)
D)
E)

Question 17

The equation of a surface is given in spherical coordinates. Give the equation of the surface in rectangular coordinates.

Question 18

The equation of a surface is given in rectangular coordinates. Give the equation of the surface in cylindrical coordinates.

Question 19

Convert from cylindrical to rectangular coordinates.

A)
B)
C)
D)
E)

Question 20

Convert from spherical to cylindrical coordinates.

A)
B)
C)
D)
E)

Question 21

Identify the quadratic surface defined by .

A) Ellipsoid
B) Hyperboloid of two sheets
C) Hyperboloid of one sheet
D) Hyperbolic paraboloid
E) Sphere

Question 22

Identify the trace of the surface defined by when .

A) Circle
B) Ellipse
C) Parabola
D) Hyperbola
E) Line

Question 23

Identify the quadratic surface defined by .

A) Ellipsoid
B) Elliptic cone
C) Elliptic paraboloid
D) Hyperbolic paraboloid
E) Not a surface

Question 24

Identify the trace of the surface defined by when .

A) Circle
B) Ellipse
C) Parabola
D) Hyperbola
E) Line

Question 25

Identify the quadratic surface defined by .

A) Ellipsoid
B) Elliptic cone
C) Elliptic paraboloid
D) Hyperbolic paraboloid
E) Sphere

Question 26

Identify the trace of the surface defined by when .

A) Circle
B) Ellipse
C) Parabola
D) Hyperbola
E) Line

Question 27

Identify the trace of the surface defined by when .

A) Circle
B) Ellipse
C) Parabola
D) Hyperbola
E) Line

Question 28

Identify the quadratic surface defined by .

Question 29

Identify the quadratic surface defined by .

A) Ellipsoid
B) Elliptic cone
C) Elliptic paraboloid
D) Hyperbolic paraboloid
E) Sphere

Question 30

Identify the trace of the surface when

Question 31

The equation of a surface is given in rectangular coordinates. Give the equation of the surface in cylindrical coordinates.

Question 32

Identify the trace of the surface defined by when .

A) Circle
B) Ellipse
C) Parabola
D) Hyperbola
E) Line

Question 33

Identify the trace of the surface defined by when .

A) Circle
B) Ellipse
C) Parabola
D) Hyperbola
E) Line

Question 34

The equation of a surface is given in rectangular coordinates. Give the equation of the surface in cylindrical coordinates.

Question 35

Identify the quadratic surface defined by .

A) Ellipsoid
B) Elliptic cone
C) Elliptic paraboloid
D) Hyperbolic paraboloid
E) Sphere

Question 36

Identify the trace of the surface defined by when .

A) Circle
B) Ellipse
C) Parabola
D) Hyperbola
E) Line

Question 37

Identify the trace of the surface defined by when .

A) Circle
B) Ellipse
C) Parabola
D) Hyperbola
E) Line

Question 38

Identify the quadratic surface defined by .

A) Ellipsoid
B) Elliptic cone
C) Elliptic paraboloid
D) Hyperbolic paraboloid
E) Sphere

Question 39

Identify the trace of the surface defined by when .

A) Circle
B) Ellipse
C) Parabola
D) Hyperbola
E) There is no trace.

Question 40

Identify the quadratic surface defined by .

A) Ellipsoid
B) Elliptic cone
C) Elliptic paraboloid
D) Hyperbolic paraboloid
E) Sphere

Question 41

Find the equation of the plane that contains and and is parallel to the line of intersection of the planes and .

Question 42

Find the acute angle of intersection of and (Round answer to the nearest degree.)

A)
B)
C)
D)
E)

Question 43

Find the acute angle of intersection of and (Round answer to the nearest degree.)

A)
B)
C)
D)
E) Parallel

Question 44

Lines and are

A) intersecting at one point
B) skew
C) parallel
D) perpendicular
E) the same line

Question 45

Identify the quadratic surface defined by .

Question 46

At what point does the line intersect the plane containing the point (-1, 0, 0) which is parallel to the xy-plane?

A) (-1, 8, 17)
B) (-1, 0,1)
C) (0, -6, 17)
D) (-1, -6, 17)
E) (2, 8, 1)

Question 47

Find an equation of the plane that passes through <4, 5, 9> and has the vector as a normal vector.

Question 48

The distance between the point (5, 6, 4) and is

A) 4
B) 3
C) -4
D) -3
E)

Question 49

Identify the quadratic surface defined by .

Question 50

Find the approximate distance between the point and .

A)
B)
C)
D)
E)

Question 51

Find the direction cosines of a vector perpendicular to .

Question 52

Lines and are

A) intersecting at one point
B) skew
C) parallel
D) perpendicular
E) the same line

Question 53

Find the equation of the plane that passes through the point and has as a normal.

A)
B)
C)
D)
E)

Question 54

At what point does the line intersect the plane containing the point (0, -1, 0) which is parallel to the xz- plane?

A) (8, 0, 5)
B) (8, -1, 5)
C) (8, 1, 1)
D) (6, 6, 5)
E) (2, 5, 1)

Question 55

A plane is perpendicular to the line given by , , and . What are the components of the normal to the plane?

A)
B)
C)
D)
E)

Question 56

Identify the quadratic surface defined by .

Question 57

Identify the quadratic surface defined by .

Question 58

At what point does the line intersect the plane containing the point (0, 0, 1) which is parallel to the xy-plane?

A) (4, -6, 1)
B) (5, 0,1)
C) (-13, 6, 1)
D) (-14, -5, 1)
E) (2, -6, 1)

Question 59

Lines and are

A) intersecting at one point, but not perpendicular
B) skew
C) parallel
D) perpendicular
E) the same line

Question 60

Find an equation of the plane that passes through <4, 5, 3> and has the vector as a normal vector.

Question 61

Find the parametric equations for the line whose vector form is given by .

A)
B)
C)
D)
E)

Question 62

Find the point of intersection of the two lines L_1:

and L₂:

Question 63

Find the parametric equations for the line passing through and .

Question 64

Find the point of intersection of the lines

A) (9, 0, 63)
B)
C) (2, -8, 7)
D) (0, 0,0)
E) (1, 1, 1)

Question 65

Find the point of intersection of the two lines L_1:
and L₂:

A)
B)
C)
D)
E)

Question 66

Find the point of intersection of the two lines L_1:

and L₂:

Question 67

Find the parametric equations for the line passing through and .

Question 68

Find the approximate distance between the point and . Approximate your answer to 4 decimal places.

Question 69

Find the direction vector for the line joining the points and .

A)
B)
C)
D)
E)

Question 70

Find the vector equation of the line given the parametric form is , , and .

A)
,

B) ,


C) ,
,

D) ,
,

E) ,
,

Question 71

Find the parametric equations for the line through the point parallel to the vector

A)
B)
C)
D) , z = 1
E)

Question 72

Find the equation of the plane through (5, 5, 11) and perpendicular to

Question 73

Find the parametric equations for the line tangent to the circle at the point .

A)
B)
C)
D)
E)

Question 74

Find the vector equation of the line though the point that is parallel to .

A) ,


B) ,
,

C) ,
,

D) ,
,

E) ,
,

Question 75

Find the parametric equations for the line whose vector form is given by .

A)
B)
C)
D) , z = 1
E)

Question 76

Find the vector equation of the line joining the points and .

A)
,

B) ,


C) ,
,

D) ,
,

E) ,
,

Question 77

Find the direction vector for the line joining the points and .

A)
B)
C)
D)
E)

Question 78

Find the vector equation of the line given the parametric form is , , and .

A)
,

B) ,


C) ,
,

D) ,
,

E) ,
,

Question 79

Find the parametric equations for the line passing through and .

Question 80

Find the distance between the point and . Approximate your answer to 4 decimal places.