Question 1

Find the length of the vector PQ where P = (1, -2, 4) and Q = (3, 4, 3).

Accepted Answer

A)    
B)  2   
C)    
D)    
E)    
A)    
B)  2   
C)    
D)    
E)

Question 2

Find the two unit vectors orthogonal to both a = 3j + 2k and b = - i - 2k.

Accepted Answer

A)  -   i -   j +   k and   i +   j -   k 
B)  k and -k 
C)  -   i +   j and   i -   j 
D)  -   i +   j +   k and   i -   j -   k 
E)  -6i - 2j + 3k and 6i + 2j - 3k 
A)  -   i -   j +   k and   i +   j -   k 
B)  k and -k 
C)  -   i +   j and   i -   j 
D)  -   i +   j +   k and   i -   j -   k 
E)  -6i - 2j + 3k and 6i + 2j - 3k

Question 3

Find the distance between (2, -1, -2) and the origin.

Accepted Answer

A)    
B)    
C)  3 
D)  4 
E)  9 
A)    
B)    
C)  3 
D)  4 
E)  9

Question 4

Given that   =   and   =   , find   .

Accepted Answer

A)    
B)    
C)    
D)    
E)  The matrix   is not invertible. 
A)    
B)    
C)    
D)    
E)  The matrix   is not invertible.

Question 5

Find the angle$\theta$ between the vectors i + 2j + 3k and 2i - 3j - k.

Accepted Answer

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

Question 6

If A is a k × m matrix and B is an m × n matrix, then BA is the transpose of AB.

Accepted Answer

A) True 
 B)False

Question 7

Find the values of a constant c such that the vector n = cj + k is normal to a family of parallel planes that intersect the elliptic cylinder x2 + 4y2 = 4 in circles.

Accepted Answer

A)  c = ±   
B)  c = ±   
C)  c = ± 2 
D)  c = ± 1 
E)  c = 0 
A)  c = ±   
B)  c = ±   
C)  c = ± 2 
D)  c = ± 1 
E)  c = 0

Question 8

Find the distance between (-2, 3, 1) and (4, 1, -3).

Accepted Answer

A)    
B)  2   
C)    
D)    
E)  2   
A)    
B)  2   
C)    
D)    
E)  2

Question 9

Describe the set of points in 3-space satisfying z2 = x2 + y2 and z = 2x.

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 10

Describe the graph of x2 + 4y2 + 16z2 = 2x - 8y.

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 11

Find the inverse of the matrix A =   .

Accepted Answer

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

Question 12

Find the equation of a plane containing the point (2, -4, 3) and the line   =   = z + 2.

Accepted Answer

A)  19x + 14y - z = -21 
B)  19x - 14y - z = 91 
C)  19x - 14y + z = 97 
D)  19x + 14y + z = -15 
E)  19x + 14y - 2z = 91 
A)  19x + 14y - z = -21 
B)  19x - 14y - z = 91 
C)  19x - 14y + z = 97 
D)  19x + 14y + z = -15 
E)  19x + 14y - 2z = 91

Question 13

If v = -i + 7j and w = 4i - 4j, then find v + w.

Accepted Answer

A)  4i + 11j 
B)  3i + 11j 
C)  3i + 3j 
D)  4i + 3j 
E)  5i + 3j 
A)  4i + 11j 
B)  3i + 11j 
C)  3i + 3j 
D)  4i + 3j 
E)  5i + 3j

Question 14

Find an equation of the sphere with centre in the xz-plane and passing through the points(0, 8, 0), (4, 6, 2), and (0, 12, 4).

Accepted Answer

A)  x2 + y2 + z2 - 14x + 24z = 257 
B)  x2 + y2 + z2 + 14x - 24z = 257 
C)  x2 + y2 + z2 - 14x + 24z = 64 
D)  x2 + y2 + z2 + 14x - 24z = 64 
E)  x2 + y2 + z2 - 14x - 24z = 257 
A)  x2 + y2 + z2 - 14x + 24z = 257 
B)  x2 + y2 + z2 + 14x - 24z = 257 
C)  x2 + y2 + z2 - 14x + 24z = 64 
D)  x2 + y2 + z2 + 14x - 24z = 64 
E)  x2 + y2 + z2 - 14x - 24z = 257

Question 15

Find an equation describing all points that are equidistant from the points A(-3, 0, 4) and B(2, 1, 5). What does this equation describe geometrically?

Accepted Answer

A)  10x + 2y + 2z = 5, the plane that right bisects the line segment AB 
B)  10x - 2y + 2z = 5, the plane that right bisects the line segment AB 
C)  10x - 2y + 18z = 5, the plane that right bisects the line segment AB 
D)  10x + 2y + 18z = 5, the plane that right bisects the line segment AB 
E)  10x + 2y + 2z = 5, the plane that contains the points A and B 
A)  10x + 2y + 2z = 5, the plane that right bisects the line segment AB 
B)  10x - 2y + 2z = 5, the plane that right bisects the line segment AB 
C)  10x - 2y + 18z = 5, the plane that right bisects the line segment AB 
D)  10x + 2y + 18z = 5, the plane that right bisects the line segment AB 
E)  10x + 2y + 2z = 5, the plane that contains the points A and B

Question 16

Consider the straight line L:   =   =   
(i) Find the point on the line L closest to the point P (-2, -1, 3).
(ii) Find the shortest distance from the point P to the line L.

Accepted Answer

(i) The point on line L closes

Question 17

Given u = i - 2j + k and v = 3i + j - 2k, find each of the following:
(a) u x v, (b) v x u, and (c) v x v.

Accepted Answer

(a) 3i + 5

Question 18

Use spherical coordinates [ R,  , $\theta$] to describe the region enclosed by the half-cone z =   and the sphere   +   +   = 4.

Accepted Answer

A)  -2 $\le$ R $\le$ 2, 0 $\le$   $\le$   , 0 $\le$ $\theta$ $\le$ 2$\pi$ 
B)  0 $\le$ R $\le$ 2, 0 $\le$   $\le$$\pi$, 0 $\le$$\theta$ $\le$ 2 $\pi$ 
C)  0 $\le$ R $\le$ 2, 0$\le$   $\le$   , 0 $\le$$\theta$ $\le$ 2$\pi$ 
D)  0 $\le$ R $\le$ 2, 0 $\le$   $\le$   , 0 $\le$ $\theta$ $\le$ 2$\pi$ 
E)  -2 $\le$ R $\le$ 2, -   $\le$   $\le$   , 0$\le$ $\theta$ $\le$ 2$\pi$ 
A)  -2 $\le$ R $\le$ 2, 0 $\le$   $\le$   , 0 $\le$ $\theta$ $\le$ 2$\pi$ 
B)  0 $\le$ R $\le$ 2, 0 $\le$   $\le$$\pi$, 0 $\le$$\theta$ $\le$ 2 $\pi$ 
C)  0 $\le$ R $\le$ 2, 0$\le$   $\le$   , 0 $\le$$\theta$ $\le$ 2$\pi$ 
D)  0 $\le$ R $\le$ 2, 0 $\le$   $\le$   , 0 $\le$ $\theta$ $\le$ 2$\pi$ 
E)  -2 $\le$ R $\le$ 2, -   $\le$   $\le$   , 0$\le$ $\theta$ $\le$ 2$\pi$

Question 19

If u × v = 0 (the zero vector), then either u = 0 or v = 0.

Accepted Answer

A) True 
 B)False

Find the length of the vector PQ where P = (1, -2, 4) and Q = (3, 4, 3).

Find the two unit vectors orthogonal to both a = 3j + 2k and b = - i - 2k.

Find the distance between (2, -1, -2) and the origin.

Given that = and = , find .

Find the angle $\theta$ between the vectors i + 2j + 3k and 2i - 3j - k.

If A is a k × m matrix and B is an m × n matrix, then BA is the transpose of AB.

Find the values of a constant c such that the vector n = cj + k is normal to a family of parallel planes that intersect the elliptic cylinder x² + 4y² = 4 in circles.

Find the distance between (-2, 3, 1) and (4, 1, -3).

Describe the set of points in 3-space satisfying z² = x² + y² and z = 2x.

Describe the graph of x² + 4y² + 16z² = 2x - 8y.

Find the inverse of the matrix A = .

Find the equation of a plane containing the point (2, -4, 3) and the line = = z + 2.

If v = -i + 7j and w = 4i - 4j, then find v + w.

Find an equation of the sphere with centre in the xz-plane and passing through the points(0, 8, 0), (4, 6, 2), and (0, 12, 4).

Find an equation describing all points that are equidistant from the points A(-3, 0, 4) and B(2, 1, 5). What does this equation describe geometrically?

Consider the straight line L: = = (i) Find the point on the line L closest to the point P (-2, -1, 3). (ii) Find the shortest distance from the point P to the line L.

Given u = i - 2j + k and v = 3i + j - 2k, find each of the following: (a) u x v, (b) v x u, and (c) v x v.

If u × v = 0 (the zero vector), then either u = 0 or v = 0.

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

Find the length of the vector PQ where P = (1, -2, 4) and Q = (3, 4, 3).

Find the two unit vectors orthogonal to both a = 3j + 2k and b = - i - 2k.

Find the distance between (2, -1, -2) and the origin.

Given that = and = , find .

Find the angle θ\thetaθ between the vectors i + 2j + 3k and 2i - 3j - k.

If A is a k × m matrix and B is an m × n matrix, then BA is the transpose of AB.

Find the values of a constant c such that the vector n = cj + k is normal to a family of parallel planes that intersect the elliptic cylinder x2 + 4y2 = 4 in circles.

Find the distance between (-2, 3, 1) and (4, 1, -3).

Describe the set of points in 3-space satisfying z2 = x2 + y2 and z = 2x.

Describe the graph of x2 + 4y2 + 16z2 = 2x - 8y.

Find the inverse of the matrix A = .

Find the equation of a plane containing the point (2, -4, 3) and the line = = z + 2.

If v = -i + 7j and w = 4i - 4j, then find v + w.

Find an equation of the sphere with centre in the xz-plane and passing through the points(0, 8, 0), (4, 6, 2), and (0, 12, 4).

Find an equation describing all points that are equidistant from the points A(-3, 0, 4) and B(2, 1, 5). What does this equation describe geometrically?

Consider the straight line L: = = (i) Find the point on the line L closest to the point P (-2, -1, 3). (ii) Find the shortest distance from the point P to the line L.

Given u = i - 2j + k and v = 3i + j - 2k, find each of the following: (a) u x v, (b) v x u, and (c) v x v.

Use spherical coordinates [ R, , θ\thetaθ ] to describe the region enclosed by the half-cone z = and the sphere + + = 4.

If u × v = 0 (the zero vector), then either u = 0 or v = 0.

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

Find the angle $\theta$ between the vectors i + 2j + 3k and 2i - 3j - k.

Find the values of a constant c such that the vector n = cj + k is normal to a family of parallel planes that intersect the elliptic cylinder x² + 4y² = 4 in circles.

Describe the set of points in 3-space satisfying z² = x² + y² and z = 2x.

Describe the graph of x² + 4y² + 16z² = 2x - 8y.