Question 1

Describe the set of points in 3-space defined by the equations x2 + y2 + z2 - 8z = 9, x = 3.

Accepted Answer

A)  a circle of radius 3 lying in the plane x = 3 and having centre at (3, 0, 4) 
B)  a circle of radius 3 lying in the plane x = 3 and having centre at (3, 0, -4) 
C)  a circle of radius 4 lying in the plane x = 3 and having centre at (3, 0, -4) 
D)  a circle of radius 4 lying in the plane x = 3 and having centre at (3, 0, 4) 
E)  a sphere of radius 5 having centre at (3, 0, 4) 
A)  a circle of radius 3 lying in the plane x = 3 and having centre at (3, 0, 4) 
B)  a circle of radius 3 lying in the plane x = 3 and having centre at (3, 0, -4) 
C)  a circle of radius 4 lying in the plane x = 3 and having centre at (3, 0, -4) 
D)  a circle of radius 4 lying in the plane x = 3 and having centre at (3, 0, 4) 
E)  a sphere of radius 5 having centre at (3, 0, 4)

Question 2

Describe the set of points in 3-space satisfying xy = 1 and z = x + y.

Accepted Answer

A)  two straight lines 
B)  one straight line 
C)  an ellipse 
D)  a parabola 
E)  a hyperbola
F) none of the above 
A)  two straight lines 
B)  one straight line 
C)  an ellipse 
D)  a parabola 
E)  a hyperbola
F) none of the above

Question 3

Find the distance between the lines   =   =   and   =   =   .

Accepted Answer

A)    units 
B)  3   units 
C)  2 units 
D)  3 units 
E)  1 unit 
A)    units 
B)  3   units 
C)  2 units 
D)  3 units 
E)  1 unit

Question 4

Given that the 3 × 3 matrix A has inverse   =   , solve the system of equations Ax = b where b has components   .

Accepted Answer

A)  x =   
B)  x =   
C)  x =   
D)  x =   
E)  x =   
A)  x =   
B)  x =   
C)  x =   
D)  x =   
E)  x =

Question 5

Express the area A of the cross-section of the ellipsoid x2 +   +   = 1 in the planez = c as a function of c where -3  $\le$ c $\le$ 3. (The area of an ellipse with semi-axes a and b is $\pi$ab.)

Accepted Answer

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

Question 6

Find two one-parameter families of straight lines that lie on the hyperbolic paraboloid z = xy.

Accepted Answer

Family I:

Question 7

Find the cartesian equation of the surface represented in spherical coordinates (R, φ, θ) by the equation R = sin(φ)   . Name and describe the surface.(Note: To describe the surface, you may list centre, vertex, axis of symmetry, etc.)

Accepted Answer

Cartesian equation : @#IMG-DLM& + @#IMG-DLM& + z² =

Question 8

Use Maple's LinearSolve routine (or other equivalent software) to solve the system of equations:u

Accepted Answer

A)  (u, v, x, y, z) = (1, 0, -1, 3, 2) 
B)  (u, v, x, y, z) = (1, 1, -1, 0, 2) 
C)  (u, v, x, y, z) = (-1, 1, 1, 3, 3) 
D)  (u, v, x, y, z) = (1, 0, -1, 3, 3) 
E)  (u, v, x, y, z) = (1, -1, 1, 3, 0) 
A)  (u, v, x, y, z) = (1, 0, -1, 3, 2) 
B)  (u, v, x, y, z) = (1, 1, -1, 0, 2) 
C)  (u, v, x, y, z) = (-1, 1, 1, 3, 3) 
D)  (u, v, x, y, z) = (1, 0, -1, 3, 3) 
E)  (u, v, x, y, z) = (1, -1, 1, 3, 0)

Question 9

Let A =   and B =   . Find BA.

Accepted Answer

The answer of Let A =   and B...

Question 10

Find the spherical coordinate equation of the surface x2 + y2 + z2 = 2z.

Accepted Answer

A)  R = 1 
B)  z =   
C)  R = 2cos($\theta$ ) 
D)    - 2R = cos(  ) 
E)  R = 2cos(  ) 
A)  R = 1 
B)  z =   
C)  R = 2cos($\theta$ ) 
D)    - 2R = cos(  ) 
E)  R = 2cos(  )

Question 11

Calculate u × v where u = 2i + j - k and v = -3i + 4j + k.

Accepted Answer

A)  5i + j - 11k 
B)  5i - j + 11k 
C)  5i + j + 11k 
D)  -5i + j + 11k 
E)  -5i + j - 11k 
A)  5i + j - 11k 
B)  5i - j + 11k 
C)  5i + j + 11k 
D)  -5i + j + 11k 
E)  -5i + j - 11k

Question 12

Find the volume of the tetrahedron spanned by the vectors u =   , v =   , and w =   .

Accepted Answer

A)  2 cubic units 
B)  3 cubic units 
C)  4 cubic units 
D)  5 cubic units 
E)  1 cubic unit 
A)  2 cubic units 
B)  3 cubic units 
C)  4 cubic units 
D)  5 cubic units 
E)  1 cubic unit

Question 13

Find the vector of unit length in the direction opposite to that of v = 4i + 7j - 4k.

Accepted Answer

A)    i +   j -   k 
B)    i +   j -   k 
C)  -   i -   j +   k 
D)  -   i -   j +   k 
E)  -   i -   j +   k 
A)    i +   j -   k 
B)    i +   j -   k 
C)  -   i -   j +   k 
D)  -   i -   j +   k 
E)  -   i -   j +   k

Question 14

Find the distance between the points (2, 4, 7) and the (1, 5, 10).

Accepted Answer

A)  3 
B)    
C)    
D)    
E)  11 
A)  3 
B)    
C)    
D)    
E)  11

Question 15

A vector in 100-dimensional Euclidean space R100 makes equal acute angles with the positive directions of the 100 coordinate axes. Approximately what is that angle?

Accepted Answer

A)  68.71° 
B)  84.26° 
C)  75.44° 
D)  87.18° 
E)  45.00° 
A)  68.71° 
B)  84.26° 
C)  75.44° 
D)  87.18° 
E)  45.00°

Question 16

Write standard form equations for the line through the point (1, 2, -6) and parallel to the vector   .

Accepted Answer

A)    =   =   
B)    =   =   
C)    =   =   
D)    =   =   
E)    =   = -   
A)    =   =   
B)    =   =   
C)    =   =   
D)    =   =   
E)    =   = -

Question 17

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

Accepted Answer

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

Question 18

Evaluate the determinant of the matrix   .

Accepted Answer

A)  60 
B)  -24 
C)  36 
D)  -18 
E)  0 
A)  60 
B)  -24 
C)  36 
D)  -18 
E)  0

Question 19

If u, v, and w are vectors in 3-space and u × v = u × w, then v = w.

Accepted Answer

A) True 
 B)False

Question 20

Use Maple's LinearSolve routine (or other equivalent software) to solve the system of equations:x

Accepted Answer

A)  (x, y, z, u, v) = (1 + t, 1 + t, 2 - t, 2 - t, t) for any real number t 
B)  (x, y, z, u, v) = (1 + t, 1 + t, 2 + t, 2 - t, 3t) for any real number t 
C)  (x, y, z, u, v) = (1 + t, 1 - t, 2 + t, 2 - t, t) for any real number t 
D)  (x, y, z, u, v) = (2 + t, 1 - t, 2 + t, 1 - t, t) for any real number t 
E)  (x, y, z, u, v) = (1 + t, 1 - t, 2 +t, 2 - t, 0) for any real number t 
A)  (x, y, z, u, v) = (1 + t, 1 + t, 2 - t, 2 - t, t) for any real number t 
B)  (x, y, z, u, v) = (1 + t, 1 + t, 2 + t, 2 - t, 3t) for any real number t 
C)  (x, y, z, u, v) = (1 + t, 1 - t, 2 + t, 2 - t, t) for any real number t 
D)  (x, y, z, u, v) = (2 + t, 1 - t, 2 + t, 1 - t, t) for any real number t 
E)  (x, y, z, u, v) = (1 + t, 1 - t, 2 +t, 2 - t, 0) for any real number t

Describe the set of points in 3-space defined by the equations x² + y² + z² - 8z = 9, x = 3.

Describe the set of points in 3-space satisfying xy = 1 and z = x + y.

Find the distance between the lines = = and = = .

Given that the 3 × 3 matrix A has inverse = , solve the system of equations Ax = b where b has components .

Find two one-parameter families of straight lines that lie on the hyperbolic paraboloid z = xy.

Find the cartesian equation of the surface represented in spherical coordinates (R, φ, θ) by the equation R = sin(φ) . Name and describe the surface.(Note: To describe the surface, you may list centre, vertex, axis of symmetry, etc.)

Use Maple's LinearSolve routine (or other equivalent software) to solve the system of equations:u

Let A = and B = . Find BA.

Find the spherical coordinate equation of the surface x² + y² + z² = 2z.

Calculate u × v where u = 2i + j - k and v = -3i + 4j + k.

Find the volume of the tetrahedron spanned by the vectors u = , v = , and w = .

Find the vector of unit length in the direction opposite to that of v = 4i + 7j - 4k.

Find the distance between the points (2, 4, 7) and the (1, 5, 10).

A vector in 100-dimensional Euclidean space R¹⁰⁰ makes equal acute angles with the positive directions of the 100 coordinate axes. Approximately what is that angle?

Write standard form equations for the line through the point (1, 2, -6) and parallel to the vector .

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

Evaluate the determinant of the matrix .

If u, v, and w are vectors in 3-space and u × v = u × w, then v = w.

Use Maple's LinearSolve routine (or other equivalent software) to solve the system of equations:x

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

Exam 11: Vectors and Coordinate Geometry in 3-Space

Describe the set of points in 3-space defined by the equations x2 + y2 + z2 - 8z = 9, x = 3.

Describe the set of points in 3-space satisfying xy = 1 and z = x + y.

Find the distance between the lines = = and = = .

Given that the 3 × 3 matrix A has inverse = , solve the system of equations Ax = b where b has components .

Express the area A of the cross-section of the ellipsoid x2 + + = 1 in the planez = c as a function of c where -3 ≤\le≤ c ≤\le≤ 3. (The area of an ellipse with semi-axes a and b is π\piπ ab.)

Find two one-parameter families of straight lines that lie on the hyperbolic paraboloid z = xy.

Find the cartesian equation of the surface represented in spherical coordinates (R, φ, θ) by the equation R = sin(φ) . Name and describe the surface.(Note: To describe the surface, you may list centre, vertex, axis of symmetry, etc.)

Use Maple's LinearSolve routine (or other equivalent software) to solve the system of equations:u

Let A = and B = . Find BA.

Find the spherical coordinate equation of the surface x2 + y2 + z2 = 2z.

Calculate u × v where u = 2i + j - k and v = -3i + 4j + k.

Find the volume of the tetrahedron spanned by the vectors u = , v = , and w = .

Find the vector of unit length in the direction opposite to that of v = 4i + 7j - 4k.

Find the distance between the points (2, 4, 7) and the (1, 5, 10).

A vector in 100-dimensional Euclidean space R100 makes equal acute angles with the positive directions of the 100 coordinate axes. Approximately what is that angle?

Write standard form equations for the line through the point (1, 2, -6) and parallel to the vector .

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

Evaluate the determinant of the matrix .

If u, v, and w are vectors in 3-space and u × v = u × w, then v = w.

Use Maple's LinearSolve routine (or other equivalent software) to solve the system of equations:x

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 set of points in 3-space defined by the equations x² + y² + z² - 8z = 9, x = 3.

Find the spherical coordinate equation of the surface x² + y² + z² = 2z.

A vector in 100-dimensional Euclidean space R¹⁰⁰ makes equal acute angles with the positive directions of the 100 coordinate axes. Approximately what is that angle?