Question 1

Find the acute angle between the planes with equations 
x + y -   z = 3 and x + y +   z = 5.

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 2

Describe the intersection of the graphs of x2 + z2 = 4 and y = 2.

Accepted Answer

A)  circle of radius 4 with centre at (0, 2, 0) in the plane z = 2 
B)  circle of radius 2 with centre at (0, 2, 0) in the plane y = 2 
C)  circle of radius 2 with centre at (0, -2, 0) in the plane y = -2 
D)  circle of radius 2 with centre at (1, 2, 1) in the plane y = 2 
E)  circle of radius   with centre at (0, 2, 0) in the plane y = 2 
A)  circle of radius 4 with centre at (0, 2, 0) in the plane z = 2 
B)  circle of radius 2 with centre at (0, 2, 0) in the plane y = 2 
C)  circle of radius 2 with centre at (0, -2, 0) in the plane y = -2 
D)  circle of radius 2 with centre at (1, 2, 1) in the plane y = 2 
E)  circle of radius   with centre at (0, 2, 0) in the plane y = 2

Question 3

Find the components of the unit vector in the same direction as v =   .

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 4

If a point P has spherical coordinates [1,  , $\theta$] and cylindrical coordinates [r,   , r], then   .

Accepted Answer

A) True 
 B)False

Question 5

Let A =   and B =   . Find AB.

Accepted Answer

A)  AB =   
B)  AB =   
C)  AB =   
D)  AB =   
E)  AB =   
A)  AB =   
B)  AB =   
C)  AB =   
D)  AB =   
E)  AB =

Question 6

Describe the graph of 4x2 - y2 = 2x + 3y.

Accepted Answer

A)  a hyperboloid of one sheet 
B)  a hyperboloid of two sheets 
C)  a hyperbolic paraboloid 
D)  an elliptic paraboloid 
E)  an ellipsoid (or a sphere)
F) a cylinder (circular, elliptic, parabolic, or hyperbolic)
G) a cone (circular, elliptic, parabolic, or hyperbolic)
H) none of the above 
A)  a hyperboloid of one sheet 
B)  a hyperboloid of two sheets 
C)  a hyperbolic paraboloid 
D)  an elliptic paraboloid 
E)  an ellipsoid (or a sphere)
F) a cylinder (circular, elliptic, parabolic, or hyperbolic)
G) a cone (circular, elliptic, parabolic, or hyperbolic)
H) none of the above

Question 7

Let A =   and B =   . Find BA.

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 8

Describe the 3-space graph of the equation x2 + y2 = 0.

Accepted Answer

A)  the xy-plane 
B)  the z-axis 
C)  a circle 
D)  a circular cylinder 
E)  a parabolic cylinder 
A)  the xy-plane 
B)  the z-axis 
C)  a circle 
D)  a circular cylinder 
E)  a parabolic cylinder

Question 9

Convert the cylindrical coordinates [2   ,   , - 2] to Cartesian coordinates and to spherical coordinates.

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 10

Describe the 3-space graph of the equation x2 + z2 = 4.

Accepted Answer

A)  a circle of radius 2 in the xz-plane having centre at the origin 
B)  a circular cylinder of radius 2 with central axis along the y-axis 
C)  a sphere of radius 2 having centre at the origin 
D)  a disk of radius 2 in the xz-plane having centre at the origin. 
E)  a circular cylinder of radius 4 with central axis along the x-axis 
A)  a circle of radius 2 in the xz-plane having centre at the origin 
B)  a circular cylinder of radius 2 with central axis along the y-axis 
C)  a sphere of radius 2 having centre at the origin 
D)  a disk of radius 2 in the xz-plane having centre at the origin. 
E)  a circular cylinder of radius 4 with central axis along the x-axis

Question 11

Use Maple or other computer algebra software to calculate the determinant of the matrix A =   .

Accepted Answer

A)  12 
B)  41 
C)  2 
D)  6 
E)  57 
A)  12 
B)  41 
C)  2 
D)  6 
E)  57

Question 12

Classify the symmetric matrix A =   as positive or negative definite, positive or negative semidefinite, or indefinite.

Accepted Answer

A)  positive definite 
B)  negative definite 
C)  positive semidefinite 
D)  negative semidefinite 
E)  indefinite 
A)  positive definite 
B)  negative definite 
C)  positive semidefinite 
D)  negative semidefinite 
E)  indefinite

Question 13

Find all values of the real number k such that the matrix A =   is invertible.

Accepted Answer

A)  k $\neq$12 , - 2 
B)  k = -4 , - 6 
C)  k = -2 , 12 
D)  k $\neq$ - 12 , 2 
E)  k = -12 , 2 
A)  k $\neq$12 , - 2 
B)  k = -4 , - 6 
C)  k = -2 , 12 
D)  k $\neq$ - 12 , 2 
E)  k = -12 , 2

Question 14

Find a value c for which A = 2i - j + 4k and B = i + cj + 8k will be perpendicular.

Accepted Answer

A)  c = -18 
B)  c = -12 
C)  c = -34 
D)  c = 34 
E)  0 
A)  c = -18 
B)  c = -12 
C)  c = -34 
D)  c = 34 
E)  0

Question 15

Find the area of a triangle that has vertices (4, 3, 6), (-2, 0, 8), (1, 5, 0).

Accepted Answer

A)    square units 
B)  49 square units 
C)    square units 
D)    square units 
E)  28 square units 
A)    square units 
B)  49 square units 
C)    square units 
D)    square units 
E)  28 square units

Question 16

Find an equation of the cylinder generated by lines through the curve y2 = x - y in the xy-plane that are parallel to the vector 2i + 2j + k.

Accepted Answer

A)    = x - y 
B)    = 2x - y 
C)    = x - 2y 
D)    = x - y 
E)    = 2x - y 
A)    = x - y 
B)    = 2x - y 
C)    = x - 2y 
D)    = x - y 
E)    = 2x - y

Question 17

Find the distance between the points (x, y, z) and (2, 0, -3).

Accepted Answer

A)    
B)    +   +   
C)    
D)    +   +   
E)    
A)    
B)    +   +   
C)    
D)    +   +   
E)

Question 18

Which of the following equations is an equation of a hyperboloid of one sheet?

Accepted Answer

A)  x2 + y2 - z2 = 0 
B)  x2 - y2 - z2 + 1= 0 
C)  x2 + y2 - z2 + 1= 0 
D)  z2 = x2 - y2 
E)  2x2 - y + 3z2 = 1 
A)  x2 + y2 - z2 = 0 
B)  x2 - y2 - z2 + 1= 0 
C)  x2 + y2 - z2 + 1= 0 
D)  z2 = x2 - y2 
E)  2x2 - y + 3z2 = 1

Question 19

Evaluate the determinant of the matrix   .

Accepted Answer

A)  4 
B)  -4 
C)  -5 
D)  -6 
E)  5 
A)  4 
B)  -4 
C)  -5 
D)  -6 
E)  5

Question 20

Use cylindrical coordinates [ r, $\theta$, z] to describe the region enclosed by the half-cone z = 2   and the paraboloid z = 8 - x2 -   .

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Find the acute angle between the planes with equations x + y - z = 3 and x + y + z = 5.

Describe the intersection of the graphs of x² + z² = 4 and y = 2.

Find the components of the unit vector in the same direction as v = .

Let A = and B = . Find AB.

Describe the graph of 4x² - y² = 2x + 3y.

Let A = and B = . Find BA.

Describe the 3-space graph of the equation x² + y² = 0.

Convert the cylindrical coordinates [2 , , - 2] to Cartesian coordinates and to spherical coordinates.

Describe the 3-space graph of the equation x² + z² = 4.

Use Maple or other computer algebra software to calculate the determinant of the matrix A = .

Classify the symmetric matrix A = as positive or negative definite, positive or negative semidefinite, or indefinite.

Find all values of the real number k such that the matrix A = is invertible.

Find a value c for which A = 2i - j + 4k and B = i + cj + 8k will be perpendicular.

Find the area of a triangle that has vertices (4, 3, 6), (-2, 0, 8), (1, 5, 0).

Find an equation of the cylinder generated by lines through the curve y² = x - y in the xy-plane that are parallel to the vector 2i + 2j + k.

Find the distance between the points (x, y, z) and (2, 0, -3).

Which of the following equations is an equation of a hyperboloid of one sheet?

Evaluate the determinant of the matrix .

Preliminaries

Limits and Continuity

Differentiation

Transcendental Functions

More Applications of Differentiation

Integration

Techniques of Integration

Applications of Integration

Conics, Parametric Curves, and Polar Curves

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Partial Differentiation

Applications of Partial Derivatives

Multiple Integration

Vector Fields

Vector Calculus

Differential Forms and Exterior Calculus

Ordinary Differential Equations

Filters

Exam 11: Vectors and Coordinate Geometry in 3-Space

Find the acute angle between the planes with equations x + y - z = 3 and x + y + z = 5.

Describe the intersection of the graphs of x2 + z2 = 4 and y = 2.

Find the components of the unit vector in the same direction as v = .

If a point P has spherical coordinates [1, , θ\thetaθ ] and cylindrical coordinates [r, , r], then .

Let A = and B = . Find AB.

Describe the graph of 4x2 - y2 = 2x + 3y.

Let A = and B = . Find BA.

Describe the 3-space graph of the equation x2 + y2 = 0.

Convert the cylindrical coordinates [2 , , - 2] to Cartesian coordinates and to spherical coordinates.

Describe the 3-space graph of the equation x2 + z2 = 4.

Use Maple or other computer algebra software to calculate the determinant of the matrix A = .

Classify the symmetric matrix A = as positive or negative definite, positive or negative semidefinite, or indefinite.

Find all values of the real number k such that the matrix A = is invertible.

Find a value c for which A = 2i - j + 4k and B = i + cj + 8k will be perpendicular.

Find the area of a triangle that has vertices (4, 3, 6), (-2, 0, 8), (1, 5, 0).

Find an equation of the cylinder generated by lines through the curve y2 = x - y in the xy-plane that are parallel to the vector 2i + 2j + k.

Find the distance between the points (x, y, z) and (2, 0, -3).

Which of the following equations is an equation of a hyperboloid of one sheet?

Evaluate the determinant of the matrix .

Use cylindrical coordinates [ r, θ\thetaθ , z] to describe the region enclosed by the half-cone z = 2 and the paraboloid z = 8 - x2 - .

Preliminaries

Limits and Continuity

Differentiation

Transcendental Functions

More Applications of Differentiation

Integration

Techniques of Integration

Applications of Integration

Conics, Parametric Curves, and Polar Curves

Sequences, Series, and Power Series

Vector Functions and Curves

Partial Differentiation

Applications of Partial Derivatives

Multiple Integration

Vector Fields

Vector Calculus

Differential Forms and Exterior Calculus

Ordinary Differential Equations

Filters

Describe the intersection of the graphs of x² + z² = 4 and y = 2.

Describe the graph of 4x² - y² = 2x + 3y.

Describe the 3-space graph of the equation x² + y² = 0.

Describe the 3-space graph of the equation x² + z² = 4.

Find an equation of the cylinder generated by lines through the curve y² = x - y in the xy-plane that are parallel to the vector 2i + 2j + k.