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693 Chapter 11: Vectors and the Geometry of Space
____ 9. Find the direction cosines of the vector u given below.
a.
b.
c.
d.
e.
____ 10. Find the direction angles of the vector u given below. Round your answer to one
decimal place.
a.
b.
c.
d.
e.
____
11.
Given
and
, find the projection of u onto v.
a.
b.
c.
d.
e.
____
12.
Given
and
, find the vector component of
u
orthogonal to
v
.
a.
b.
c.
d.
e.
11.3 The Dot Product of Two Vectors
694
____ 13.
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.
$12,079.58
$12,118.58
$24,138.16
$12,068.58
$24,136.16
____ 14. Suppose a 58000-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.
29,000.00 lb
28,615.00 lb
29,135.00 lb
29,056.00 lb
29,011.00 lb
695
Chapter 11: Vectors and the Geometry of Space
____
15.
Suppose a 48000-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 perpendicular to the hill.
Round your answer to one decimal place.
47,008.8 lb
47,426.8 lb
47,328.8 lb
47,270.8 lb
47,289.8 lb
____ 16.
An object is pulled 5 feet horizontally across a floor, using a force of 95 pounds. The
direction of the force is
above the horizontal. Find the work done. Round your answer to two
decimal places.
467.78 ft-lb
398.56 ft-lb
82.48 ft-lb
6.96 ft-lb
511.94 ft-lb
11.3 The Dot Product of Two Vectors
696
____ 17.
Suppose a toy wagon is pulled by exerting a force of 25 pounds on a handle that
makes a
angle with the horizontal as shown in the figure below. Find the work done in pulling
the wagon 50 feet. Round your answer to one decimal place.
1,200.0 ft-lb
427.5 ft-lb
2,350.2 ft-lb
1,174.6 ft-lb
2,348.2 ft-lb
____ 18. Suppose a car is towed using a force of 1300 newtons. The chain used to pull the car
makes a angle with the horizontal. Find the work done in towing the car 4 kilometers. Round
your answer to two decimal places.
9,426.60 km-N
4,712.80 km-N
4,726.80 km-N
9,424.60 km-N
2,197.61 km-N
697 Chapter 11: Vectors and the Geometry of Space
11.3 The Dot Product of Two Vectors
Answer Section
11.4 The Cross Product of Two Vectors in Space
698
11.4 The Cross Product of Two Vectors in Space
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1.
Find the cross product of the unit vector
.
a.
b.
c.
d.
e.
____
2.
Sketch the graph of the cross product of the unit vector
.
a.
d.
b. e.
699 Chapter 11: Vectors and the Geometry of Space
c.
____ 3. Given , find .
a.
b.
c.
d.
e.
____ 4. Given and find .
a.
b.
c.
d.
e.
____ 5. Given and find .
a.
b.
c.
d.
e.
11.4 The Cross Product of Two Vectors in Space
700
____
6.
Given
and
, find
.
a.
b.
c.
d.
e.
____
7.
Given
and
, find
.
a.
b.
c.
d.
e.
____
8.
Find a unit vector that is orthogonal to both
and
.
a.
b.
c.
d.
e.
____ 9. Find the area of the parallelogram that has the given vectors and as
adjacent sides.
a.
b.
c.
d.
e.
701
Chapter 11: Vectors and the Geometry of Space
____
10.
Find the area of a parallelogram that has the given vectors as adjacent sides. Round
your answer to two decimal places.
a.
b.
c.
d.
e.
____ 11. Find the area of the parallelogram with the vertices
, and . Round your answer to two decimal places.
35.00
3.00
5.92
7.00
2.65
____ 12. Find the area of the triangle with the vertices and
. Round your answer to two decimal places.
a.
20.00
b.
8.00
c.
6.00
d.
2.24
e.
4.47
____ 13.
A child applies the brakes on a bicycle by applying a downward force of 25 pounds
on the pedal when the crank makes a
angle with the horizontal (see figure). The crank is 6 inches
in length. Find the torque at P. Round your answer to two decimal places.
11.4 The Cross Product of Two Vectors in Space
702
12.50 ft-lb
6.25 ft-lb
75.00 ft-lb
10.83 ft-lb
21.65 ft-lb
____ 14. 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.
2,598.08 ft-lb
259.81 ft-lb
212.13 ft-lb
519.62 ft-lb
150.00 ft-lb
____ 15. Find the triple scalar product .
a.
b.
c.
d.
e.
703
Chapter 11: Vectors and the Geometry of Space
____
16.
Use the triple scalar product to find the volume of the parallelepiped having adjacent
edges given by the vectors
516
____ 17. Find the volume of the parallelepiped with the following vertices.
a.
b.
c.
d.
e.
11.4 The Cross Product of Two Vectors in Space
704
11.4 The Cross Product of Two Vectors in Space
Answer Section
705 Chapter 11: Vectors and the Geometry of Space
11.5 Lines and Planes in Space
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find a set of parametric equations of the line through the point parallel to the
vector .
a.
b.
c.
d.
e.
____ 2. Find a set of symmetric equations of the line through the point parallel to the
vector .
a.
b.
c.
d.
e.
____ 3. Find the set of parametric equations of the line through the point that is parallel
to the line , ,and .
a.
b.
c.
d.
e.
____ 4. Find a set of parametric equations of the line through the points and
.
a.
b.
c.
d.
e.
11.5 Lines and Planes in Space
706
____ 5.
Find a set of symmetric equations of the line through the points
and
.
a.
b.
c.
d.
e.
____ 6. Find a set of parametric equations of the line that passes through the point
and is parallel to the xy-plane and yz-plane.
a.
b.
c.
d.
e.
____ 7. Determine whether any of the lines given below are parallel or identical.
are identical
b. are identical
are identical
are parallel
are parallel
707 Chapter 11: Vectors and the Geometry of Space
____ 8. Determine whether any of the lines given below are parallel or identical.
are parallel
are identical
c. are parallel
are identical
are parallel
____ 9. Determine whether the lines given below intersect, and, if so, find the point of
intersection.
The lines intersect at .
The lines are parallel.
c.
The lines intersect at
.
d.
The lines intersect at
.
e.
The lines are not parallel but do not intersect.
____ 10.
Determine whether the lines given below intersect, and, if so, find the point of
intersection.
a.
The lines are parallel.
b.
The lines intersect at
.
c.
The lines intersect at
.
d.
The lines intersect at
.
e.
The lines are not parallel but do not intersect.
11.5 Lines and Planes in Space
708
____
11.
Find an equation of a plane passing through the given point and perpendicular to the
given vector.
Point:
Vector:
a.
b.
c.
d.
e.
____
12.
Find an equation of a plane passing through the following three points.
a.
b.
c.
d.
e.
____ 13. Find an equation of a plane passing through the points
and perpendicular to the plane
.
a.
b.
c.
d.
e.
____ 14. Determine whether the following planes are parallel, orthogonal, or neither. If they are
neither parallel nor orthogonal, find the angle of intersection.
The planes are neither parallel nor orthogonal, the angle of intersection is .
The planes are parallel.
The planes are neither parallel nor orthogonal, the angle of intersection is .
The planes are orthogonal.
709
Chapter 11: Vectors and the Geometry of Space
____
15.
Determine whether the following planes are parallel, orthogonal, or neither. If they
are neither parallel nor orthogonal, find the angle of intersection.
The planes are neither parallel nor orthogonal, the angle of intersection is .
The planes are neither parallel nor orthogonal, the angle of intersection is .
The planes are orthogonal.
The planes are parallel.
____
16.
Find the distance between the point
and the plane
.
a.
b.
c.
d.
e.
____
17.
Find the distance between the planes given below.
a.
b.
c.
d.
e.
11.5 Lines and Planes in Space
710
____ 18.
Find the distance between the point
and the line given by the set of
parametric equations
.
5 22 /11
2
3 110 /11
4
1
____ 19. A chute at the top of a grain elevator of a combine funnels the grain into a bin (see
figure). Find the angle between two adjacent sides. Round your answer to one decimal place.
a.
b.
c.
d.
e.
711
Chapter 11: Vectors and the Geometry of Space
____
20.
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.
Find the distance between the two insects at time . Round your answer to two decimal places.
inches
inches
inches
inches
inches
____ 21.
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.
inches
inches
inches
inches
inches
