Question 1

Find the magnitude of u × v and the unit vector parallel to u × v in the direction of u × v.
-u = 2i + 2j - k, v = -i + k

Accepted Answer

A)  3;   i +   j -   k 
B)  9;   i +   j -   k 
C)  3;   i -   j +   k 
D)  9;   i -   j +   k 
A)  3;   i +   j -   k 
B)  9;   i +   j -   k 
C)  3;   i -   j +   k 
D)  9;   i -   j +   k

Question 2

Match the equation with the surface it defines.    
--   +   +   = 1

Accepted Answer

A)  Figure 1 
B)  Figure 3 
C)  Figure 4 
D)  Figure 2 
A)  Figure 1 
B)  Figure 3 
C)  Figure 4 
D)  Figure 2

Question 3

Solve the problem.
-For the triangle with vertices located at A( 3, 3, 5), B( 4, 2, 2), and C(1, 1, 1) , find a vector from vertex C to the midpoint of side AB.

Accepted Answer

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

Question 4

Identify the surface by name.
-  +   -   = 0

Accepted Answer

A)  Hyperboloid of one sheet 
B)  Hyperbolic cylinder 
C)  Elliptic cone 
D)  Ellipsoid 
A)  Hyperboloid of one sheet 
B)  Hyperbolic cylinder 
C)  Elliptic cone 
D)  Ellipsoid

Question 5

Write the equation for the plane.
-The plane passing through the points (1, 2, 6), (-1, 4, 0), and (3, 1, 5).

Accepted Answer

A)  9x - 2y + 8z = 53 
B)  4x + 7y + z = 24 
C)  2x + 4y + 5z = 40 
D)  6x + 11y - z = 22 
A)  9x - 2y + 8z = 53 
B)  4x + 7y + z = 24 
C)  2x + 4y + 5z = 40 
D)  6x + 11y - z = 22

Question 6

Determine whether the pairs of lines are parallel, intersect at a single point, or are skew. If the lines are parallel, determine whether they are the same line (and thus intersect at all points). If the lines intersect at a single point, determine the point of intersection.
-r = (3, 1, 4) + t(-1, 6, -2); R = (-6, 55, -14) + t(5, -30, 10)

Accepted Answer

A)  The lines are skew. 
B)  The lines intersect at a single point,   
C)  The lines are parallel, distinct lines. 
D)  The lines are the same line. 
A)  The lines are skew. 
B)  The lines intersect at a single point,   
C)  The lines are parallel, distinct lines. 
D)  The lines are the same line.

Question 7

Find v ∙ u.
-v =   and u =

Accepted Answer

A)    i -   j 
B)    i 
C)    
D)  0 
A)    i -   j 
B)    i 
C)    
D)  0

Question 8

Express the vector in the form ai + bj + ck.
-  if A is the point ( -4, -8, -5) and B is the point ( 1, -15, -2)

Accepted Answer

A)  v = 5i + 7j - 3k 
B)  v = 5i - 7j + 3k 
C)  v = 5i - 7j - 3k 
D)  v = 5i + 7j + 3k 
A)  v = 5i + 7j - 3k 
B)  v = 5i - 7j + 3k 
C)  v = 5i - 7j - 3k 
D)  v = 5i + 7j + 3k

Question 9

Find the angle between u and v in radians.
-u = -2i - 2j, v = 4i + 9j + 8k

Accepted Answer

A)  1.59 
B)  1.73 
C)  -0.81 
D)  2.38 
A)  1.59 
B)  1.73 
C)  -0.81 
D)  2.38

Question 10

Use a calculator to find the acute angle between the planes to the nearest thousandth of a radian.
-4x - 3y - 5z = 7 and- 8x + 2y - 10z = -9

Accepted Answer

A)  1.528 rad 
B)  0.131 rad 
C)  0.593 rad 
D)  1.439 rad 
A)  1.528 rad 
B)  0.131 rad 
C)  0.593 rad 
D)  1.439 rad

Question 11

Find an equation for the sphere with the given center and radius.
-Center (-2, 0, 0), radius = 4

Accepted Answer

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

Question 12

Find the magnitude of u × v and the unit vector parallel to u × v in the direction of u × v.
-u = -3i - 2j - 3k, v = 6i + 4j + 6k

Accepted Answer

A)    ;   ( 3 i + 2 j + 3 k ) 
B)  0; no direction 
C)  0; 3i+ 2j + 3k 
D)    ;   ( 3 i + 2 j + 3 k ) 
A)    ;   ( 3 i + 2 j + 3 k ) 
B)  0; no direction 
C)  0; 3i+ 2j + 3k 
D)    ;   ( 3 i + 2 j + 3 k )

Question 13

Find the triple scalar product (u x v) ∙ w of the given vectors.
-u = 2i - 4j + 3k; v = -5i - 7j + 8k; w = 9i - 2j + 4k

Accepted Answer

A)  563 
B)  -39 
C)  153 
D)  -173 
A)  563 
B)  -39 
C)  153 
D)  -173

Question 14

Give a geometric description of the set of points whose coordinates satisfy the given conditions.
-  +   +   > 1

Accepted Answer

A)  All points outside the sphere of radius 1 
B)  All points in space 
C)  All points outside the cylinder with radius 1 
D)  All points on the surface of the cylinder with radius 1 
A)  All points outside the sphere of radius 1 
B)  All points in space 
C)  All points outside the cylinder with radius 1 
D)  All points on the surface of the cylinder with radius 1

Question 15

Solve the problem.
-Find the magnitude of the torque in foot-pounds at point P for the following lever:

Accepted Answer

A)      ft-lb 
B)      ft-lb 
C)  -     ft-lb 
D)      ft-lb 
A)      ft-lb 
B)      ft-lb 
C)  -     ft-lb 
D)      ft-lb

Question 16

Solve the problem.
-An airplane is flying in the direction 50° west of north at 817 km/hr. Find the component form of the velocity of the airplane, assuming that the positive x-axis represents due east and the positive y-axis represents due north.

Accepted Answer

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

Question 17

Use a calculator to find the acute angle between the planes to the nearest thousandth of a radian.
--3x - 2y - 10z = 1 and 6x + 8y - 10z = 9

Accepted Answer

A)  1.471 rad 
B)  1.116 rad 
C)  0.825 rad 
D)  0.455 rad 
A)  1.471 rad 
B)  1.116 rad 
C)  0.825 rad 
D)  0.455 rad

Question 18

Find an equation for the sphere with the given center and radius.
-Center (0, -10, -7), radius = 5

Accepted Answer

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

Question 19

Find an equation for the sphere with the given center and radius.
-Center (-8, 10, 0), radius = 5

Accepted Answer

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

Question 20

Solve the problem.
-A bird flies from its nest 4 km in the direction 4° north of east, where it stops to rest on a tree. It then flies 7 km in the direction 4° south of west and lands atop a telephone pole. With an xy-coordinate system where the origin is the bird's nest, the x-axis points east, and the y-axis points north, at what point is the telephone pole located?

Accepted Answer

A)  ( -6.983, -0.4883) 
B)  ( 3.990, 0.2790) 
C)  ( -2.993, -0.2093) 
D)  ( 1.961, 1.548) 
A)  ( -6.983, -0.4883) 
B)  ( 3.990, 0.2790) 
C)  ( -2.993, -0.2093) 
D)  ( 1.961, 1.548)

Find the magnitude of u × v and the unit vector parallel to u × v in the direction of u × v. -u = 2i + 2j - k, v = -i + k

Match the equation with the surface it defines. -- + + = 1

Solve the problem. -For the triangle with vertices located at A( 3, 3, 5), B( 4, 2, 2), and C(1, 1, 1) , find a vector from vertex C to the midpoint of side AB.

Identify the surface by name. - + - = 0

Write the equation for the plane. -The plane passing through the points (1, 2, 6), (-1, 4, 0), and (3, 1, 5).

Find v ∙ u. -v = and u =

Express the vector in the form ai + bj + ck. - if A is the point ( -4, -8, -5) and B is the point ( 1, -15, -2)

Find the angle between u and v in radians. -u = -2i - 2j, v = 4i + 9j + 8k

Use a calculator to find the acute angle between the planes to the nearest thousandth of a radian. -4x - 3y - 5z = 7 and- 8x + 2y - 10z = -9

Find an equation for the sphere with the given center and radius. -Center (-2, 0, 0), radius = 4

Find the magnitude of u × v and the unit vector parallel to u × v in the direction of u × v. -u = -3i - 2j - 3k, v = 6i + 4j + 6k

Find the triple scalar product (u x v) ∙ w of the given vectors. -u = 2i - 4j + 3k; v = -5i - 7j + 8k; w = 9i - 2j + 4k

Give a geometric description of the set of points whose coordinates satisfy the given conditions. - + + > 1

Solve the problem. -Find the magnitude of the torque in foot-pounds at point P for the following lever:

Solve the problem. -An airplane is flying in the direction 50° west of north at 817 km/hr. Find the component form of the velocity of the airplane, assuming that the positive x-axis represents due east and the positive y-axis represents due north.

Use a calculator to find the acute angle between the planes to the nearest thousandth of a radian. --3x - 2y - 10z = 1 and 6x + 8y - 10z = 9

Find an equation for the sphere with the given center and radius. -Center (0, -10, -7), radius = 5

Find an equation for the sphere with the given center and radius. -Center (-8, 10, 0), radius = 5

Functions

Limits

Derivatives

Applications of the Derivative

Integration

Applications of Integration

Logarithmic, Exponential, and Hyperbolic Functions

Integration Techniques

Differential Equations

Sequences and Infinite Series

Power Series

Parametric and Polar Curves

Vector-Valued Functions

Functions of Several Variables

Multiple Integration

Vector Calculus

Filters

Exam 13: Vectors and the Geometry of Space

Find the magnitude of u × v and the unit vector parallel to u × v in the direction of u × v. -u = 2i + 2j - k, v = -i + k

Match the equation with the surface it defines. -- + + = 1

Solve the problem. -For the triangle with vertices located at A( 3, 3, 5), B( 4, 2, 2), and C(1, 1, 1) , find a vector from vertex C to the midpoint of side AB.

Identify the surface by name. - + - = 0

Write the equation for the plane. -The plane passing through the points (1, 2, 6), (-1, 4, 0), and (3, 1, 5).

Find v ∙ u. -v = and u =

Express the vector in the form ai + bj + ck. - if A is the point ( -4, -8, -5) and B is the point ( 1, -15, -2)

Find the angle between u and v in radians. -u = -2i - 2j, v = 4i + 9j + 8k

Use a calculator to find the acute angle between the planes to the nearest thousandth of a radian. -4x - 3y - 5z = 7 and- 8x + 2y - 10z = -9

Find an equation for the sphere with the given center and radius. -Center (-2, 0, 0), radius = 4

Find the magnitude of u × v and the unit vector parallel to u × v in the direction of u × v. -u = -3i - 2j - 3k, v = 6i + 4j + 6k

Find the triple scalar product (u x v) ∙ w of the given vectors. -u = 2i - 4j + 3k; v = -5i - 7j + 8k; w = 9i - 2j + 4k

Give a geometric description of the set of points whose coordinates satisfy the given conditions. - + + > 1

Solve the problem. -Find the magnitude of the torque in foot-pounds at point P for the following lever:

Solve the problem. -An airplane is flying in the direction 50° west of north at 817 km/hr. Find the component form of the velocity of the airplane, assuming that the positive x-axis represents due east and the positive y-axis represents due north.

Use a calculator to find the acute angle between the planes to the nearest thousandth of a radian. --3x - 2y - 10z = 1 and 6x + 8y - 10z = 9

Find an equation for the sphere with the given center and radius. -Center (0, -10, -7), radius = 5

Find an equation for the sphere with the given center and radius. -Center (-8, 10, 0), radius = 5

Functions

Limits

Derivatives

Applications of the Derivative

Integration

Applications of Integration

Logarithmic, Exponential, and Hyperbolic Functions

Integration Techniques

Differential Equations

Sequences and Infinite Series

Power Series

Parametric and Polar Curves

Vector-Valued Functions

Functions of Several Variables

Multiple Integration

Vector Calculus

Filters