Question 1

Find a unit vector that is orthogonal to both   and   .

Accepted Answer

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

Question 2

Identify the equation (written in terms of cylindrical or spherical coordinates) of the following graph. ​   ​

Accepted Answer

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

Question 3

Given   and   find   .

Accepted Answer

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

Question 4

To carry a 100-pound cylindrical weight, two workers lift on the ends of short ropes tied to an eyelet on the top center of the cylinder. One rope makes a   angle away from the vertical and the other makes a   angle as shown in the figure below. Find each rope's tension if the resultant force is vertical.Round your answer to two decimal places. ​   ​

Accepted Answer

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

Question 5

Find an equation in rectangular coordinates for the equation given in spherical coordinates. Express the coefficients to two decimal places.

Accepted Answer

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

Question 6

Find the magnitude of the vector given below.

Accepted Answer

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

Question 7

Convert the following point from spherical coordinates to cylindrical coordinates.

Accepted Answer

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

Question 8

Graph the solid described in spherical coordinates. ​   ​

Accepted Answer

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

Question 9

Suppose the vector   gives the numbers of hamburgers, chicken sandwiches, and cheeseburgers, respectively, sold at a fast-food restaurant in one week. The vector   gives the prices (in dollars) per unit for the three food items. Determine the total revenue that the restaurant earned on its three products for that week. Round your answer to two decimal places.

Accepted Answer

A)  $12,296.28 
B)  $12,321.28 
C)  $24,557.56 
D)  $12,278.28 
E)  $24,555.56 
A)  $12,296.28 
B)  $12,321.28 
C)  $24,557.56 
D)  $12,278.28 
E)  $24,555.56

Question 10

Both the magnitude and the direction of the force on a crankshaft change as the crankshaft rotates. Find the torque on the crankshaft using the position and data shown in the figure. Round your answer to two decimal places.

Accepted Answer

A)  2,771.28 ft-lb 
B)  277.13 ft-lb 
C)  226.27 ft-lb 
D)  554.26 ft-lb 
E)  160.00 ft-lb 
A)  2,771.28 ft-lb 
B)  277.13 ft-lb 
C)  226.27 ft-lb 
D)  554.26 ft-lb 
E)  160.00 ft-lb

Question 11

Given the equation of a surface of revolution   , find the equation of its generating curve in the xy-plane rotating about the x-axis.

Accepted Answer

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

Question 12

Identify the following quadric surface.

Accepted Answer

A)  hyperboloid of two sheets 
B)  elliptic cone 
C)  ellipsoid 
D)  elliptic paraboloid 
E)  hyperboloid of one sheet 
A)  hyperboloid of two sheets 
B)  elliptic cone 
C)  ellipsoid 
D)  elliptic paraboloid 
E)  hyperboloid of one sheet

Question 13

Find the distance between the points given below. ​
(3, 1, 4 ), ( 6, 6, 7 )
​

Accepted Answer

A)    or 6.5574 to four decimal places. 
B)    or 6.4807 to four decimal places. 
C)    or 6.6332 to four decimal places. 
D)    or 6.0828 to four decimal places. 
E)    or 6.7082 to four decimal places. 
A)    or 6.5574 to four decimal places. 
B)    or 6.4807 to four decimal places. 
C)    or 6.6332 to four decimal places. 
D)    or 6.0828 to four decimal places. 
E)    or 6.7082 to four decimal places.

Question 14

Find the angle between the vectors for u and v.

Accepted Answer

A)  46.88   
B)  23.20   
C)  66.80   
D)  42.59   
E)  88.85   
A)  46.88   
B)  23.20   
C)  66.80   
D)  42.59   
E)  88.85

Question 15

Find a set of symmetric equations of the line through the points   and   .

Accepted Answer

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

Question 16

Find the direction angles of the vector u given below. Round your answer to one decimal place.

Accepted Answer

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

Question 17

Suppose a 48,000-pound truck is parked on a   slope as shown in the figure. Assume the only force to overcome is that due to gravity. Find the force required to keep the truck from rolling down the hill. Round your answer to two decimal places. ​   ​

Accepted Answer

A)  8,335.11 lb 
B)  7,971.11 lb 
C)  8,472.11 lb 
D)  8,384.11 lb 
E)  8,348.11 lb 
A)  8,335.11 lb 
B)  7,971.11 lb 
C)  8,472.11 lb 
D)  8,384.11 lb 
E)  8,348.11 lb

Question 18

Find the unit vector in the direction of u.

Accepted Answer

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

Question 19

Find the standard equation of a sphere that has diameter with the end points given below. (4, 8, 3), (2, 4, 7)

Accepted Answer

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

Question 20

Find the standard equation of the sphere with center (4, 1, 1), and radius 2.

Accepted Answer

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

Find a unit vector that is orthogonal to both and .

Identify the equation (written in terms of cylindrical or spherical coordinates) of the following graph.

Given and find .

Find an equation in rectangular coordinates for the equation given in spherical coordinates. Express the coefficients to two decimal places.

Find the magnitude of the vector given below.

Convert the following point from spherical coordinates to cylindrical coordinates.

Graph the solid described in spherical coordinates.

Both the magnitude and the direction of the force on a crankshaft change as the crankshaft rotates. Find the torque on the crankshaft using the position and data shown in the figure. Round your answer to two decimal places.

Given the equation of a surface of revolution , find the equation of its generating curve in the xy-plane rotating about the x-axis.

Identify the following quadric surface.

Find the distance between the points given below. (3, 1, 4 ), ( 6, 6, 7 )

Find the angle between the vectors for u and v.

Find a set of symmetric equations of the line through the points and .

Find the direction angles of the vector u given below. Round your answer to one decimal place.

Suppose a 48,000-pound truck is parked on a slope as shown in the figure. Assume the only force to overcome is that due to gravity. Find the force required to keep the truck from rolling down the hill. Round your answer to two decimal places.

Find the unit vector in the direction of u.

Find the standard equation of a sphere that has diameter with the end points given below. (4, 8, 3), (2, 4, 7)

Find the standard equation of the sphere with center (4, 1, 1), and radius 2.

Preparation for Calculus

Limits and Their Properties

Differentiation

Applications of Differentiation

Integration

Differential Equations

Applications of Integration

Integration Techniques and Improper Integrals

Infinite Series

Conics, Parametric Equations, and Polar Coordinates

Vector-Valued Functions

Functions of Several Variables

Multiple Integration

Vector Anal

Filters

Exam 11: Vectors and the Geometry of Space

Find a unit vector that is orthogonal to both and .

Identify the equation (written in terms of cylindrical or spherical coordinates) of the following graph. ​ ​

Given and find .

Find an equation in rectangular coordinates for the equation given in spherical coordinates. Express the coefficients to two decimal places.

Find the magnitude of the vector given below.

Convert the following point from spherical coordinates to cylindrical coordinates.

Graph the solid described in spherical coordinates. ​ ​

Both the magnitude and the direction of the force on a crankshaft change as the crankshaft rotates. Find the torque on the crankshaft using the position and data shown in the figure. Round your answer to two decimal places.

Given the equation of a surface of revolution , find the equation of its generating curve in the xy-plane rotating about the x-axis.

Identify the following quadric surface.

Find the distance between the points given below. ​ (3, 1, 4 ), ( 6, 6, 7 ) ​

Find the angle between the vectors for u and v.

Find a set of symmetric equations of the line through the points and .

Find the direction angles of the vector u given below. Round your answer to one decimal place.

Suppose a 48,000-pound truck is parked on a slope as shown in the figure. Assume the only force to overcome is that due to gravity. Find the force required to keep the truck from rolling down the hill. Round your answer to two decimal places. ​ ​

Find the unit vector in the direction of u.

Find the standard equation of a sphere that has diameter with the end points given below. (4, 8, 3), (2, 4, 7)

Find the standard equation of the sphere with center (4, 1, 1), and radius 2.

Preparation for Calculus

Limits and Their Properties

Differentiation

Applications of Differentiation

Integration

Differential Equations

Applications of Integration

Integration Techniques and Improper Integrals

Infinite Series

Conics, Parametric Equations, and Polar Coordinates

Vector-Valued Functions

Functions of Several Variables

Multiple Integration

Vector Anal

Filters

Identify the equation (written in terms of cylindrical or spherical coordinates) of the following graph.

Graph the solid described in spherical coordinates.

Find the distance between the points given below. (3, 1, 4 ), ( 6, 6, 7 )

Suppose a 48,000-pound truck is parked on a slope as shown in the figure. Assume the only force to overcome is that due to gravity. Find the force required to keep the truck from rolling down the hill. Round your answer to two decimal places.