Question 1

Find two unit vectors orthogonal to the two given vectors.

Accepted Answer

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

Question 2

Find the distance between the given objects. The planes 5x - 9y + 6z = 76 and 5x - 9y + 6z = -100

Accepted Answer

A)  176 
B)    
C)  0 
D)    
A)  176 
B)    
C)  0 
D)

Question 3

Determine if the lines are parallel, skew or intersect.

Accepted Answer

A)  parallel 
B)  skew 
C)  intersect 
A)  parallel 
B)  skew 
C)  intersect

Question 4

Suppose that there are two forces acting on a barge being towed along a river. One force is exerted due south by the river current and has a magnitude of 7 units (the exact nature of the force unit is unimportant). The other force is exerted by a tugboat and has a magnitude of 9 units towards the north and 3 units towards the east. What is the net force acting on the barge?

Accepted Answer

A)  9 units towards the north, 4 units towards the east 
B)  16 units towards the north, 3 units towards the east 
C)  2 units towards the north, 3 units towards the east 
D)  3 units towards the north, 2 units towards the east 
A)  9 units towards the north, 4 units towards the east 
B)  16 units towards the north, 3 units towards the east 
C)  2 units towards the north, 3 units towards the east 
D)  3 units towards the north, 2 units towards the east

Question 5

Match the equation to its graph. [The figures below are not necessarily drawn to scale.]

Accepted Answer

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

Question 6

Find the distance from the point Q to the given line. Round to the nearest thousandth.   , line through   and

Accepted Answer

A)  17.117 
B)  1.018 
C)  17.647 
D)  14.314 
A)  17.117 
B)  1.018 
C)  17.647 
D)  14.314

Question 7

Which of the following statements are true? i.   ii.   iii.

Accepted Answer

A)  i only 
B)  ii only 
C)  i and ii only 
D)  i and iii only 
E)  iii only 
A)  i only 
B)  ii only 
C)  i and ii only 
D)  i and iii only 
E)  iii only

Question 8

Identify the surface.

Accepted Answer

A)  elliptic paraboloid 
B)  cone 
C)  hyperboloid of one sheet 
D)  hyperbolic paraboloid 
A)  elliptic paraboloid 
B)  cone 
C)  hyperboloid of one sheet 
D)  hyperbolic paraboloid

Question 9

Compute ||-2a - 3b||.

Accepted Answer

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

Question 10

Find a vector with the given magnitude and in the same direction as the given vector. Magnitude 2,

Accepted Answer

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

Question 11

Find the indicated volume. Volume of the parallelpiped with three adjacent edges formed by   ,   , and

Accepted Answer

A)  232 
B)    
C)    
D)  312 
A)  232 
B)    
C)    
D)  312

Question 12

A constant force of   pounds moves an object in a straight line from the point (0, 0) to the point (20, -19). Compute the work done.

Accepted Answer

A)  -880 
B)  0 
C)  -750 
D)  1150 
A)  -880 
B)  0 
C)  -750 
D)  1150

Question 13

Use geometry to identify the cross product.

Accepted Answer

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

Question 14

The motor of a small boat produces a speed of 2 mph in still water. The boat is travelling in flowing water with a current velocity given by   . In what direction should the boat head to travel due east? (Represent the direction with a unit vector.)

Accepted Answer

A)  i 
B)    
C)    
D)  The boat cannot travel due east against this current. 
A)  i 
B)    
C)    
D)  The boat cannot travel due east against this current.

Question 15

Which one of the following statements is always true?

Accepted Answer

A)  Two planes always intersect in a line. 
B)  The equation   can be used to calculate the distance between any two planes. 
C)  A plane cannot contain two skew lines. 
D)  The equation   describes the plane containing the point   and containing the vector   . 
A)  Two planes always intersect in a line. 
B)  The equation   can be used to calculate the distance between any two planes. 
C)  A plane cannot contain two skew lines. 
D)  The equation   describes the plane containing the point   and containing the vector   .

Question 16

Determine whether the given planes are the same. -3(x - 5) + (y - 9) - 3(z + 3) = 0
-3(x + 4) + (y + 6) - 3(z - 1) = 0

Accepted Answer

A)  The planes are the same. 
B)  The planes are not the same. 
A)  The planes are the same. 
B)  The planes are not the same.

Question 17

Find an equation of the given plane. The plane containing the points   ,   and

Accepted Answer

A)  3x - 9y - 7z = 82 
B)  9x - 13y - 10z = 137 
C)  9x - 13y - 10z = 0 
D)  58x + 64y - 31z = -185 
A)  3x - 9y - 7z = 82 
B)  9x - 13y - 10z = 137 
C)  9x - 13y - 10z = 0 
D)  58x + 64y - 31z = -185

Question 18

Identify the surface.

Accepted Answer

A)  elliptic paraboloid 
B)  hyperboloid of two sheets 
C)  hyperboloid of one sheet 
D)  hyperbolic paraboloid 
A)  elliptic paraboloid 
B)  hyperboloid of two sheets 
C)  hyperboloid of one sheet 
D)  hyperbolic paraboloid

Question 19

Assume that the ball is moving into the page (and away from you) with the indicated spin. Determine the direction of the Magnus force.

Accepted Answer

A)  to the right 
B)  to the left 
C)  up 
D)  down 
E)  The Magnus force is zero. 
A)  to the right 
B)  to the left 
C)  up 
D)  down 
E)  The Magnus force is zero.

Question 20

Sketch the given plane. 3x - y + 2z = 6

Accepted Answer

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

Find two unit vectors orthogonal to the two given vectors.

Find the distance between the given objects. The planes 5x - 9y + 6z = 76 and 5x - 9y + 6z = -100

Determine if the lines are parallel, skew or intersect.

Match the equation to its graph. [The figures below are not necessarily drawn to scale.]

Find the distance from the point Q to the given line. Round to the nearest thousandth. , line through and

Which of the following statements are true? i. ii. iii.

Identify the surface.

Compute ||-2a - 3b||.

Find a vector with the given magnitude and in the same direction as the given vector. Magnitude 2,

Find the indicated volume. Volume of the parallelpiped with three adjacent edges formed by , , and

A constant force of pounds moves an object in a straight line from the point (0, 0) to the point (20, -19). Compute the work done.

Use geometry to identify the cross product.

The motor of a small boat produces a speed of 2 mph in still water. The boat is travelling in flowing water with a current velocity given by . In what direction should the boat head to travel due east? (Represent the direction with a unit vector.)

Which one of the following statements is always true?

Determine whether the given planes are the same. -3(x - 5) + (y - 9) - 3(z + 3) = 0 -3(x + 4) + (y + 6) - 3(z - 1) = 0

Find an equation of the given plane. The plane containing the points , and

Identify the surface.

Assume that the ball is moving into the page (and away from you) with the indicated spin. Determine the direction of the Magnus force.

Sketch the given plane. 3x - y + 2z = 6

Preliminaries

Limits and Continuity

Differentiation

Applications of the Derivative

Integration

Applications of the Definite Integral

Exponentials, Logarithms and Other Transcendental Functions

Integration Techniques

First-Order Differential Equations

Infinite Series

Parametric Equations and Polar Coordinates

Vector-Valued Functions

Functions of Several Variables and Partial Differentiation

Multiple Integrals

Vector Calculus

Second Order Differential Equations

Filters

Exam 12: Vectors and the Geometry of Space

Find two unit vectors orthogonal to the two given vectors.

Find the distance between the given objects. The planes 5x - 9y + 6z = 76 and 5x - 9y + 6z = -100

Determine if the lines are parallel, skew or intersect.

Match the equation to its graph. [The figures below are not necessarily drawn to scale.]

Find the distance from the point Q to the given line. Round to the nearest thousandth. , line through and

Which of the following statements are true? i. ii. iii.

Identify the surface.

Compute ||-2a - 3b||.

Find a vector with the given magnitude and in the same direction as the given vector. Magnitude 2,

Find the indicated volume. Volume of the parallelpiped with three adjacent edges formed by , , and

A constant force of pounds moves an object in a straight line from the point (0, 0) to the point (20, -19). Compute the work done.

Use geometry to identify the cross product.

The motor of a small boat produces a speed of 2 mph in still water. The boat is travelling in flowing water with a current velocity given by . In what direction should the boat head to travel due east? (Represent the direction with a unit vector.)

Which one of the following statements is always true?

Determine whether the given planes are the same. -3(x - 5) + (y - 9) - 3(z + 3) = 0 -3(x + 4) + (y + 6) - 3(z - 1) = 0

Find an equation of the given plane. The plane containing the points , and

Identify the surface.

Assume that the ball is moving into the page (and away from you) with the indicated spin. Determine the direction of the Magnus force.

Sketch the given plane. 3x - y + 2z = 6

Preliminaries

Limits and Continuity

Differentiation

Applications of the Derivative

Integration

Applications of the Definite Integral

Exponentials, Logarithms and Other Transcendental Functions

Integration Techniques

First-Order Differential Equations

Infinite Series

Parametric Equations and Polar Coordinates

Vector-Valued Functions

Functions of Several Variables and Partial Differentiation

Multiple Integrals

Vector Calculus

Second Order Differential Equations

Filters