Question 1

Describe the set of points in 3-space satisfying z2 = 1 + x2 + 2y2 and z = 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 F

Question 2

Find the equation of a plane that contains the lines=     =   and   -   == z - 2.

Accepted Answer

A)  53x + 51y - 37z = 67 
B)  53x - 47y + 5z = -731 
C)  53x - 47y +57 = 11 
D)  53x + 51y - 37z = -67 
E)  53x - 47y - 5z = 11 
A)  53x + 51y - 37z = 67 
B)  53x - 47y + 5z = -731 
C)  53x - 47y +57 = 11 
D)  53x + 51y - 37z = -67 
E)  53x - 47y - 5z = 11 A

Question 3

A plane in 3-space is uniquely determined by any three different points that lie on it.

Accepted Answer

A) True 
 B)False  False

Question 4

Find the distance from the point P(1, -1, 2) to the line connecting the point (3, 1, 4) to (1, -3, 0).

Accepted Answer

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

Question 5

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)  (a) 3i + 5j + 7k, (b) -3i - 5j - 7k, (c) 0 
B)  (a) 3i - 5j + 7k, (b) -3i + 5j - 7k, (c) -4i +12j 
C)  (a) 3i + 5j - 5k, (b) -3i - 5j + 5k, (c) 12j 
D)  (a) 3i - 5j - 5k, (b) -3i + 5j + 5k, (c) -4 i + 6k 
E)  (a) 3i + 5j + 7k, (b) -3i - 5j - 5k, (c) 0 
A)  (a) 3i + 5j + 7k, (b) -3i - 5j - 7k, (c) 0 
B)  (a) 3i - 5j + 7k, (b) -3i + 5j - 7k, (c) -4i +12j 
C)  (a) 3i + 5j - 5k, (b) -3i - 5j + 5k, (c) 12j 
D)  (a) 3i - 5j - 5k, (b) -3i + 5j + 5k, (c) -4 i + 6k 
E)  (a) 3i + 5j + 7k, (b) -3i - 5j - 5k, (c) 0

Question 6

If u = 3i + j + 4k, v = -i + 2j, and w = - 2i - 3j + 5k, evaluate u × (v × w).

Accepted Answer

A)  -13i + 19j + 5k 
B)  -14i + 20j + 4k 
C)  -13i - 19j - 5k 
D)  14i - 20j - 4k 
E)  13i - 19j - 5k 
A)  -13i + 19j + 5k 
B)  -14i + 20j + 4k 
C)  -13i - 19j - 5k 
D)  14i - 20j - 4k 
E)  13i - 19j - 5k

Question 7

Find the equation of the line of intersection of the planes 2x + 6y - z - 19 = 0,x + 3 y - 3z = 2.

Accepted Answer

A)  r = (-17 - 3t) i + t j + (- 5 - t) k, t    R 
B)    =   =   
C)  3x - y + z = 56 
D)  r = (17 - 3t) i + t j + (5 - t) k, t    R 
E)  The two planes do not intersect. 
A)  r = (-17 - 3t) i + t j + (- 5 - t) k, t    R 
B)    =   =   
C)  3x - y + z = 56 
D)  r = (17 - 3t) i + t j + (5 - t) k, t    R 
E)  The two planes do not intersect.

Question 8

If line L passes through point (1, 2, 3) and is perpendicular to the xy-plane, what are the coordinates of the points on the line that are at a distance 7 from the point P(3, -1, 5)?

Accepted Answer

A)  (1, 2, 1) and (1, 2, 9) 
B)  (1, 2, 0) and (1, 2, 10) 
C)  (1, 2, -1) and (1, 2, 11) 
D)  (1, 2, -2) and (1, -2, 12) 
E)  (1, 2, -4) and (1, 2, 10) 
A)  (1, 2, 1) and (1, 2, 9) 
B)  (1, 2, 0) and (1, 2, 10) 
C)  (1, 2, -1) and (1, 2, 11) 
D)  (1, 2, -2) and (1, -2, 12) 
E)  (1, 2, -4) and (1, 2, 10)

Question 9

Describe the 3-space graph of the equation z2 = y2 + x2.

Accepted Answer

A)  a surface of revolution obtained by rotating the parabola z = x2 in the xz-plane about the z-axis 
B)  a circular cone with vertex at the origin, axis along the z-axis, and semi-vertical angle   
C)  a circular cylinder with axis along the z-axis radius z 
D)  a circle having radius z in the xy-plane 
E)  a surface of revolution obtained by rotating the parabola z = y2 in the yz-plane about the z-axis 
A)  a surface of revolution obtained by rotating the parabola z = x2 in the xz-plane about the z-axis 
B)  a circular cone with vertex at the origin, axis along the z-axis, and semi-vertical angle   
C)  a circular cylinder with axis along the z-axis radius z 
D)  a circle having radius z in the xy-plane 
E)  a surface of revolution obtained by rotating the parabola z = y2 in the yz-plane about the z-axis

Question 10

Evaluate the determinant of the matrix   .

Accepted Answer

A)  21 
B)  3 
C)  -3 
D)  -21 
E)  0 
A)  21 
B)  3 
C)  -3 
D)  -21 
E)  0

Question 11

Given u = 2i - j + 2k and v = i - j + 4k, find each of the following:(a) u × v, (b) v × u, and (c) v × v.

Accepted Answer

A)  (a) -2i - 6j - k, (b) 2i + 6j + k, (c) 0 
B)  (a) -2i - 6j - k, (b) -2i - 6j - k, (c) -4i +8j -4k 
C)  (a) 2i + 6j + k, (b) -2i - 6j - k, (c) -4i +8j -4k 
D)  (a) -2i - 6j + k, (b) 2i + 6j - k, (c) 0 
E)  (a) -2i + 6j - k, (b) 2i - 6j + k, (c) 0 
A)  (a) -2i - 6j - k, (b) 2i + 6j + k, (c) 0 
B)  (a) -2i - 6j - k, (b) -2i - 6j - k, (c) -4i +8j -4k 
C)  (a) 2i + 6j + k, (b) -2i - 6j - k, (c) -4i +8j -4k 
D)  (a) -2i - 6j + k, (b) 2i + 6j - k, (c) 0 
E)  (a) -2i + 6j - k, (b) 2i - 6j + k, (c) 0

Question 12

For what value of the constant k will the vectors   ,   , and   be coplanar?

Accepted Answer

A)  k = 8 
B)  k = -7 
C)  k = 6 
D)  k = -5 
E)  k = 0 
A)  k = 8 
B)  k = -7 
C)  k = 6 
D)  k = -5 
E)  k = 0

Question 13

Describe the graph of x2 + 4z2 = 2y.

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 14

Find the eigenvalues and corresponding eigenvectors of the matrix A =   .

Accepted Answer

A)  0,   ; 3,   ; 4,   
B)  0,   ; 3,   ; 4,   
C)  0,   ; 3,   ; 4,   
D)  0,   ; 3,   ; 4,   
E)  0,   ; 3,   ; 4,   
A)  0,   ; 3,   ; 4,   
B)  0,   ; 3,   ; 4,   
C)  0,   ; 3,   ; 4,   
D)  0,   ; 3,   ; 4,   
E)  0,   ; 3,   ; 4,

Question 15

If v is a vector in the xy-plane, |v| = 7   , and v makes an angle of 3 $\pi$/4 with the positive direction of the x-axis, then what are the components of v?

Accepted Answer

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

Question 16

If u = 3i + j + 4k, v = -i + 2j, and w = - 2i - 3j + 5k, evaluate u . (v × w).

Accepted Answer

A)  63 
B)  64 
C)  -64 
D)  62 
E)  -63 
A)  63 
B)  64 
C)  -64 
D)  62 
E)  -63

Question 17

Find the vector that makes equal acute angles with the positive coordinate axes and has length 2.

Accepted Answer

A)    (i + j + k) 
B)  2(i + j + k) 
C)    (i + j + k) 
D)    (i + j + k) 
E)    (i + j + k) 
A)    (i + j + k) 
B)  2(i + j + k) 
C)    (i + j + k) 
D)    (i + j + k) 
E)    (i + j + k)

Question 18

det   = det   .

Accepted Answer

A) True 
 B)False

Question 19

The planes x + 2y - 4z = 10 and -2x - 4y - 8z = 11 are parallel.

Accepted Answer

A) True 
 B)False

Question 20

Use Maple or other computer algebra software to 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

Describe the set of points in 3-space satisfying z² = 1 + x² + 2y² and z = y.

Find the equation of a plane that contains the lines= = and - == z - 2.

A plane in 3-space is uniquely determined by any three different points that lie on it.

Find the distance from the point P(1, -1, 2) to the line connecting the point (3, 1, 4) to (1, -3, 0).

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 = 3i + j + 4k, v = -i + 2j, and w = - 2i - 3j + 5k, evaluate u × (v × w).

Find the equation of the line of intersection of the planes 2x + 6y - z - 19 = 0,x + 3 y - 3z = 2.

If line L passes through point (1, 2, 3) and is perpendicular to the xy-plane, what are the coordinates of the points on the line that are at a distance 7 from the point P(3, -1, 5)?

Describe the 3-space graph of the equation z² = y² + x².

Evaluate the determinant of the matrix .

Given u = 2i - j + 2k and v = i - j + 4k, find each of the following:(a) u × v, (b) v × u, and (c) v × v.

For what value of the constant k will the vectors , , and be coplanar?

Describe the graph of x² + 4z² = 2y.

Find the eigenvalues and corresponding eigenvectors of the matrix A = .

If u = 3i + j + 4k, v = -i + 2j, and w = - 2i - 3j + 5k, evaluate u . (v × w).

Find the vector that makes equal acute angles with the positive coordinate axes and has length 2.

det = det .

The planes x + 2y - 4z = 10 and -2x - 4y - 8z = 11 are parallel.

Use Maple or other computer algebra software to classify the symmetric matrix A = as positive or negative definite, positive or negative semidefinite, or indefinite.

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 satisfying z2 = 1 + x2 + 2y2 and z = y.

Find the equation of a plane that contains the lines= = and - == z - 2.

A plane in 3-space is uniquely determined by any three different points that lie on it.

Find the distance from the point P(1, -1, 2) to the line connecting the point (3, 1, 4) to (1, -3, 0).

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 = 3i + j + 4k, v = -i + 2j, and w = - 2i - 3j + 5k, evaluate u × (v × w).

Find the equation of the line of intersection of the planes 2x + 6y - z - 19 = 0,x + 3 y - 3z = 2.

If line L passes through point (1, 2, 3) and is perpendicular to the xy-plane, what are the coordinates of the points on the line that are at a distance 7 from the point P(3, -1, 5)?

Describe the 3-space graph of the equation z2 = y2 + x2.

Evaluate the determinant of the matrix .

Given u = 2i - j + 2k and v = i - j + 4k, find each of the following:(a) u × v, (b) v × u, and (c) v × v.

For what value of the constant k will the vectors , , and be coplanar?

Describe the graph of x2 + 4z2 = 2y.

Find the eigenvalues and corresponding eigenvectors of the matrix A = .

If v is a vector in the xy-plane, |v| = 7 , and v makes an angle of 3 π\piπ /4 with the positive direction of the x-axis, then what are the components of v?

If u = 3i + j + 4k, v = -i + 2j, and w = - 2i - 3j + 5k, evaluate u . (v × w).

Find the vector that makes equal acute angles with the positive coordinate axes and has length 2.

det = det .

The planes x + 2y - 4z = 10 and -2x - 4y - 8z = 11 are parallel.

Use Maple or other computer algebra software to classify the symmetric matrix A = as positive or negative definite, positive or negative semidefinite, or indefinite.

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 satisfying z² = 1 + x² + 2y² and z = y.

Describe the 3-space graph of the equation z² = y² + x².

Describe the graph of x² + 4z² = 2y.