Question 1

Determine whether u and v are orthogonal, parallel or neither.

Accepted Answer

A)  Parallel, same direction 
B)  Orthogonal 
C)  Neither parallel nor orthogonal 
D)  Parallel, opposite direction 
A)  Parallel, same direction 
B)  Orthogonal 
C)  Neither parallel nor orthogonal 
D)  Parallel, opposite direction

Question 2

Find the area of a parallelogram that has the given vectors as adjacent sides. Round your answer to two decimal places.

Accepted Answer

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

Question 3

Find an equation of a plane passing through the points   and perpendicular to the plane   .

Accepted Answer

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

Question 4

Given   and   , find the projection of u onto v.

Accepted Answer

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

Question 5

Find the cross product of the unit vector   .

Accepted Answer

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

Question 6

Identify the following quadric surface.

Accepted Answer

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

Question 7

Find the vector v whose initial and terminal points are given below. (5,6.1), (7, 10.1)

Accepted Answer

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

Question 8

Two insects are crawling along different lines in three-space. At time t (in minutes), the first insect is at the point   where   . Also, at time t, the second insect is at the point   where   . Assume distances are given in inches. How close do the insects get? Round your answer to two decimal places.

Accepted Answer

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

Question 9

The top of a rubber bushing designed to absorb vibrations in an automobile is the surface of revolution generated by revolving the curve   in the yz-plane about the z-axis. Find an equation for the surface of revolution.

Accepted Answer

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

Question 10

Find the distance between the point   and the plane   .

Accepted Answer

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

Question 11

Suppose a plane is flying in the direction   . Its speed with respect to the air is 900 kilometers per hour. The wind at the plane's altitude is from the southwest at 100 kilometers per hour (see figure). What is the true direction of the plane? Round your answer to two decimal places. ​   ​

Accepted Answer

A)    North of West 
B)    North of West 
C)    North of East 
D)    North of West 
E)    North of South 
A)    North of West 
B)    North of West 
C)    North of East 
D)    North of West 
E)    North of South

Question 12

Find an equation in cylindrical coordinates for the equation given in rectangular coordinates.

Accepted Answer

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

Question 13

Determine whether any of the lines given below are parallel or identical.

Accepted Answer

A)    are identical 
B)    are identical 
C)    are identical 
D)    are parallel 
E)    are parallel 
A)    are identical 
B)    are identical 
C)    are identical 
D)    are parallel 
E)    are parallel

Question 14

Find a set of symmetric equations of the line through the point   parallel to the vector   .

Accepted Answer

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

Question 15

Find an equation in rectangular coordinates for the equation given in spherical coordinates. ​   ​

Accepted Answer

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

Question 16

Find the direction cosines of the vector u given below.

Accepted Answer

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

Question 17

The top of a rubber bushing designed to absorb vibrations in an automobile is the surface of revolution generated by revolving the curve   in the yz-plane about the z-axis. All measurements are in centimeters and the bushing is set on the xy-plane. Use the shell method to find its volume. Round your answer to one decimal place. ​

Accepted Answer

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

Question 18

Use the shell method to find the volume of the solid below the surface formed by revolving the curve   about the z-axis and above the xy-plane.

Accepted Answer

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

Question 19

Convert the following point from rectangular coordinates to cylindrical coordinates. Give any angles in radians. ​   ​

Accepted Answer

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

Question 20

Determine whether u and v are orthogonal, parallel and pointing in the same direction, parallel and pointing in opposite directions, or neither.

Accepted Answer

A)  Neither parallel nor orthogonal 
B)  Parallel, same direction 
C)  Orthogonal 
D)  Parallel, opposite direction 
A)  Neither parallel nor orthogonal 
B)  Parallel, same direction 
C)  Orthogonal 
D)  Parallel, opposite direction

Determine whether u and v are orthogonal, parallel or neither.

Find the area of a parallelogram that has the given vectors as adjacent sides. Round your answer to two decimal places.

Find an equation of a plane passing through the points and perpendicular to the plane .

Given and , find the projection of u onto v.

Find the cross product of the unit vector .

Identify the following quadric surface.

Find the vector v whose initial and terminal points are given below. (5,6.1), (7, 10.1)

Two insects are crawling along different lines in three-space. At time t (in minutes), the first insect is at the point where . Also, at time t, the second insect is at the point where . Assume distances are given in inches. How close do the insects get? Round your answer to two decimal places.

The top of a rubber bushing designed to absorb vibrations in an automobile is the surface of revolution generated by revolving the curve in the yz-plane about the z-axis. Find an equation for the surface of revolution.

Find the distance between the point and the plane .

Suppose a plane is flying in the direction . Its speed with respect to the air is 900 kilometers per hour. The wind at the plane's altitude is from the southwest at 100 kilometers per hour (see figure). What is the true direction of the plane? Round your answer to two decimal places.

Find an equation in cylindrical coordinates for the equation given in rectangular coordinates.

Determine whether any of the lines given below are parallel or identical.

Find a set of symmetric equations of the line through the point parallel to the vector .

Find an equation in rectangular coordinates for the equation given in spherical coordinates.

Find the direction cosines of the vector u given below.

Use the shell method to find the volume of the solid below the surface formed by revolving the curve about the z-axis and above the xy-plane.

Convert the following point from rectangular coordinates to cylindrical coordinates. Give any angles in radians.

Determine whether u and v are orthogonal, parallel and pointing in the same direction, parallel and pointing in opposite directions, or neither.

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

Determine whether u and v are orthogonal, parallel or neither.

Find the area of a parallelogram that has the given vectors as adjacent sides. Round your answer to two decimal places.

Find an equation of a plane passing through the points and perpendicular to the plane .

Given and , find the projection of u onto v.

Find the cross product of the unit vector .

Identify the following quadric surface.

Find the vector v whose initial and terminal points are given below. (5,6.1), (7, 10.1)

Two insects are crawling along different lines in three-space. At time t (in minutes), the first insect is at the point where . Also, at time t, the second insect is at the point where . Assume distances are given in inches. How close do the insects get? Round your answer to two decimal places.

The top of a rubber bushing designed to absorb vibrations in an automobile is the surface of revolution generated by revolving the curve in the yz-plane about the z-axis. Find an equation for the surface of revolution.

Find the distance between the point and the plane .

Suppose a plane is flying in the direction . Its speed with respect to the air is 900 kilometers per hour. The wind at the plane's altitude is from the southwest at 100 kilometers per hour (see figure). What is the true direction of the plane? Round your answer to two decimal places. ​ ​

Find an equation in cylindrical coordinates for the equation given in rectangular coordinates.

Determine whether any of the lines given below are parallel or identical.

Find a set of symmetric equations of the line through the point parallel to the vector .

Find an equation in rectangular coordinates for the equation given in spherical coordinates. ​ ​

Find the direction cosines of the vector u given below.

Use the shell method to find the volume of the solid below the surface formed by revolving the curve about the z-axis and above the xy-plane.

Convert the following point from rectangular coordinates to cylindrical coordinates. Give any angles in radians. ​ ​

Determine whether u and v are orthogonal, parallel and pointing in the same direction, parallel and pointing in opposite directions, or neither.

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

Suppose a plane is flying in the direction . Its speed with respect to the air is 900 kilometers per hour. The wind at the plane's altitude is from the southwest at 100 kilometers per hour (see figure). What is the true direction of the plane? Round your answer to two decimal places.

Find an equation in rectangular coordinates for the equation given in spherical coordinates.

Convert the following point from rectangular coordinates to cylindrical coordinates. Give any angles in radians.