Question 1

Find the intersection of the planes. -7x - 4y - 8z = -123 and 6x - 8y - 4z = 42

Accepted Answer

A)  x = 9 - 80t, y = -3 + 20t, z = 9 - 32t 
B)  x = 80t, y = -20t, z = 32t 
C)  x = 9 - 48t, y = -3 - 76t, z = 9 + 80t 
D)  The planes are parallel. 
A)  x = 9 - 80t, y = -3 + 20t, z = 9 - 32t 
B)  x = 80t, y = -20t, z = 32t 
C)  x = 9 - 48t, y = -3 - 76t, z = 9 + 80t 
D)  The planes are parallel.

Question 2

Prove that projc(a + b) = projc a + projc b for any non-zero vectors a, b, and c.

Accepted Answer

Question 3

Determine whether the given planes are the same. 9(x + 7) + 7(y + 2) - (z + 4) = 0
9(x + 8) + 7(y - 8) - (z + 6) = 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 4

Find two unit vectors parallel to the given vector.

Accepted Answer

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

Question 5

Find the vector with the given polar form.

Accepted Answer

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

Question 6

Compute   .

Accepted Answer

A)    
B)  17 
C)    
D)  169 
A)    
B)  17 
C)    
D)  169

Question 7

If   ,   , and   for   and   , show that the point (x, y, z) lies on the upper half of the cone   . Show all your work.

Accepted Answer

@#IMG-DLM& Since @#IMG-DLM& is positive

Question 8

Identify the surface.

Accepted Answer

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

Question 9

Sketch the given plane. y + 2z = 6

Accepted Answer

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

Question 10

The motor of a small boat produces a speed of 3 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 11

Find symmetric equations of the line through   and parallel to   .

Accepted Answer

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

Question 12

Find an equation of the plane containing the lines.

Accepted Answer

A)  18x + 25y - 41z = 174 
B)  18x - 137y - 41z = 660 
C)  -9x - 5y - 7z = -27 
D)  The lines are skew. 
A)  18x + 25y - 41z = 174 
B)  18x - 137y - 41z = 660 
C)  -9x - 5y - 7z = -27 
D)  The lines are skew.

Question 13

Compute the cross product a × b.

Accepted Answer

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

Question 14

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 15

Find projb a for the given vectors.

Accepted Answer

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

Question 16

A small communications satellite is being maneuvered by two on-board thrusters. One thruster generates a force with magnitude 13 newtons in a direction   . In response to the applied thrusts, the satellite moves under a total force of   (the components are in units of newtons). Find the thrust vector for the second thruster.

Accepted Answer

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

Question 17

A constant force of   pounds moves an object in a straight line a distance of 15 feet, and the work done is 65 ft-lb. If the motion of the object started at the point (0, 0), find the coordinates of the final position of the object. [Assume that the final position is somewhere in the first quadrant of the coordinate system.]

Accepted Answer

A)    
B)    
C)    
D)  The final position cannot be determined. 
A)    
B)    
C)    
D)  The final position cannot be determined.

Question 18

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 19

Use the cross product to determine the angle betweeen the vectors, assuming that   . Round to the nearest thousandth.

Accepted Answer

A)  1.571 
B)  0.384 
C)  0.658 
D)  0.912 
A)  1.571 
B)  0.384 
C)  0.658 
D)  0.912

Question 20

Determine whether the given lines are the same. x = 3 - 2t, y = -8 + 6t, z = 2 - t and x = 3 - 8t, y = -8 - 4t, z = 2 - 8t

Accepted Answer

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

Find the intersection of the planes. -7x - 4y - 8z = -123 and 6x - 8y - 4z = 42

Prove that proj_c(a + b) = proj_c a + proj_c b for any non-zero vectors a, b, and c.

Determine whether the given planes are the same. 9(x + 7) + 7(y + 2) - (z + 4) = 0 9(x + 8) + 7(y - 8) - (z + 6) = 0

Find two unit vectors parallel to the given vector.

Find the vector with the given polar form.

Compute .

If , , and for and , show that the point (x, y, z) lies on the upper half of the cone . Show all your work.

Identify the surface.

Sketch the given plane. y + 2z = 6

The motor of a small boat produces a speed of 3 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.)

Find symmetric equations of the line through and parallel to .

Find an equation of the plane containing the lines.

Compute the cross product a × b.

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

Find proj_ba for the given vectors.

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

Use the cross product to determine the angle betweeen the vectors, assuming that . Round to the nearest thousandth.

Determine whether the given lines are the same. x = 3 - 2t, y = -8 + 6t, z = 2 - t and x = 3 - 8t, y = -8 - 4t, z = 2 - 8t

Preliminaries

Limits and Continuity

Differentiation

Applications of the Derivative

Integration

Applications of the Definite Integral

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 11: Vectors and the Geometry of Space

Find the intersection of the planes. -7x - 4y - 8z = -123 and 6x - 8y - 4z = 42

Prove that projc(a + b) = projc a + projc b for any non-zero vectors a, b, and c.

Determine whether the given planes are the same. 9(x + 7) + 7(y + 2) - (z + 4) = 0 9(x + 8) + 7(y - 8) - (z + 6) = 0

Find two unit vectors parallel to the given vector.

Find the vector with the given polar form.

Compute .

If , , and for and , show that the point (x, y, z) lies on the upper half of the cone . Show all your work.

Identify the surface.

Sketch the given plane. y + 2z = 6

The motor of a small boat produces a speed of 3 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.)

Find symmetric equations of the line through and parallel to .

Find an equation of the plane containing the lines.

Compute the cross product a × b.

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

Find projb a for the given vectors.

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

Use the cross product to determine the angle betweeen the vectors, assuming that . Round to the nearest thousandth.

Determine whether the given lines are the same. x = 3 - 2t, y = -8 + 6t, z = 2 - t and x = 3 - 8t, y = -8 - 4t, z = 2 - 8t

Preliminaries

Limits and Continuity

Differentiation

Applications of the Derivative

Integration

Applications of the Definite Integral

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

Prove that proj_c(a + b) = proj_c a + proj_c b for any non-zero vectors a, b, and c.

Find proj_ba for the given vectors.