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673 Chapter 11: Vectors and the Geometry of Space
11.1 Vectors in the Plane
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1.
Find vectors
u
and
v
whose initial and terminal points are given. Determine whether
u and v are equivalent.
u: (6,5), (9,11)
v: (3,–5), (6,1)
a.
b.
c.
d.
e.
____
2.
Find the vector v whose initial and terminal points are given below.
(5,5), (7,2)
a.
b.
c.
d.
e.
____
3.
Find the vector v whose initial and terminal points are given below.
(5.5,2.1), (9.5, 6.1)
a.
b.
c.
d.
e.
____ 4. Given , find .
a.
b.
c.
d.
e.
11.1 Vectors in the Plane
674
____
5.
Given
and
, find
.
a.
b.
c.
d.
e.
____
6.
Given
and
, find
.
a.
b.
c.
d.
e.
____
7.
The vector v and its initial point is given. Find the terminal point.
,
initial point
a.
b.
c.
d.
e.
____
8.
Find the magnitude of the vector given below. Round your answer to four decimal
places.
a.
b.
c.
d.
e.
675 Chapter 11: Vectors and the Geometry of Space
____ 9. Find the unit vector in the direction of u.
a.
b.
c.
d.
e.
____
10.
Find the vector
v
with the given magnitude ||
v
|| = 185 and the same direction as
.
a.
11
187 , 8
187
b.
8
185 ,
11
185
c.
11
185 ,
8
185
d.
8
187 ,
8
187
2618 , 4 187
____ 11.
Find the component form of vector v given its magnitude is
and the angle it
makes with the positive x-axis is
.
a.
b.
c.
d.
e.
11.1 Vectors in the Plane
676
____ 12.
Find the component form of vector u + v given
and
and the angles
that u and v make with the positive x-axis are
and
.
a.
b.
c.
d.
e.
____
13.
Three forces with magnitudes
pounds,
pounds and
pounds act on an
object at angles
,
, and
respectively, with the positive x-axis. Find the direction and
magnitude of the resultant force. (The choices below are given to two decimal places.)
a.
with the positive x-axis
b.
with the positive x-axis
c.
with the positive x-axis
d.
with the positive x-axis
e.
pounds at an angle of
with the positive x-axis
____ 14. Suppose a gun with a muzzle velocity of 1200 feet per second is fired at an angle of
above the horizontal. Find the horizontal component of the velocity. Round your answer to two
decimal places.
ft/sec
ft/sec
ft/sec
ft/sec
ft/sec
677
Chapter 11: Vectors and the Geometry of Space
____
15.
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.
a.
b.
c.
d.
e.
11.1 Vectors in the Plane
678
____ 16.
To carry a 500-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 the vertical component of each
worker's force. Round your answer to two decimal places.
a.
b.
c.
d.
e.
679
Chapter 11: Vectors and the Geometry of Space
____
17.
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.
a.
b.
c.
d.
e.
11.1 Vectors in the Plane
680
____ 18.
Suppose a plane is flying in the direction
(relative to the west). Its speed with
respect to the air is 600 kilometers per hour. The wind at the plane's altitude is from the southwest at
80 kilometers per hour (see figure). What is its speed with respect to the ground? Round your answer
to one decimal place.
a.
b.
c.
d.
e.
____ 19. Suppose a plane flies at a constant groundspeed of 600 miles per hour due east and
encounters a 10 mile-per-hour wind from the northwest. Both the airspeed and the compass direction
must change to for the plane to maintain its groundspeed and eastward direction. Find the airspeed to
maintain its groundspeed and eastward direction. Round your answer to two decimal places.
a.
b.
c.
d.
e.
____ 20. Suppose a plane flies at a constant groundspeed of 500 miles per hour due east and
encounters a 50 mile-per-hour wind from the northwest. Both the airspeed and the compass direction
must change to for the plane to maintain its groundspeed and eastward direction. Find the airspeed to
maintain its groundspeed and eastward direction. Round your answer to two decimal places.
a.
b.
c.
d.
e.
681 Chapter 11: Vectors and the Geometry of Space
11.1 Vectors in the Plane
Answer Section
11.2 Space Coordinates and Vectors in Space
682
11.2 Space Coordinates and Vectors in Space
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find the coordinates of the point that is located 4 units behind the yz-plane, 6 units to the left
of xz-plane, and 4 units above the xy-plane.
(–4,–6,–4)
(–4,6,4)
(4,–6,4)
(–4,6,–4)
(–4,–6,4)
____ 2. Find the distance between the points given below.
(–4, 2, –3 ), ( 0, 5, 1 )
41 or 6.4031 to four decimal places.
2 10 or 6.3246 to four decimal places.
42 or 6.4807 to four decimal places.
43 or 5.9161 to four decimal places.
43 or 6.5574 to four decimal places.
____ 3. Find the coordinates of the midpoint of the line segment joining the points given
below.
(4, 8, 4), (7, 10, 7)
(5.5, 1, 5.5)
(1.5, 1, 5.5)
(1.5, 9, 5.5)
(1.5, 9, 1.5)
(5.5, 9, 5.5)
____ 4. Find the standard equation of the sphere with center (2, 8, 4), and radius 2.
a.
b.
c.
d.
e.
683
Chapter 11: Vectors and the Geometry of Space
____
5.
Find the standard equation of a sphere that has diameter with the end points given
below.
(3, –2, 4), (7, 12,
4) a.
b.
c.
d.
e.
____ 6.
Find the standard equation of the sphere with center at
and tangent to the
yz-plane.
a.
b.
c.
d.
e.
____ 7. Complete the square to write the following equation in the standard form for a
sphere.
a.
b.
c.
d.
e.
11.2 Space Coordinates and Vectors in Space
684
____ 8.
Find the component form of the vector u with the given initial and terminal points.
Initial point: (5, 7, 8 )
Terminal point: (–2 , 10 , 10 )
a.
b.
c.
d.
e.
____ 9. Find the magnitude of the vector v given its initial and terminal points. Round your answer
to four decimal places.
Initial point: (–2, –4, –5)
Terminal point: (–3, –9, –10)
a.
b.
c.
d.
e.
____
a.
b.
c.
d.
e.
____
a.
b.
c.
d.
e.
or 7.3485
or 7.4833
or
or 4.0000
or 8.0000
Given the vector v and its initial point find the terminal point of the vector.
initial point
11. Given , find .
685 Chapter 11: Vectors and the Geometry of Space
____ 12. Given , sketch the graph of .
a.
b.
c.
11.2 Space Coordinates and Vectors in Space
686
d.
e.
____ 13. Find the vector
, ,
a.
b.
c.
d.
e.
687 Chapter 11: Vectors and the Geometry of Space
____ 14. Find the magnitude of the vector given below.
a.
b.
c.
d.
e.
____ 15. Find the unit vector in the direction of u.
a.
b.
c.
d.
e.
____ 16. Find a unit vector u in the direction opposite of .
1/ 77 )
(1/ 59 )
1/ 59 )
(1/ 77 )
1/6 2 )
11.2 Space Coordinates and Vectors in Space
688
____ 17.
Find the vector v with the magnitude
and direction of
.
a.
b.
c.
d.
e.
____ 18. Given the vector v lies in the yz-plane, has magnitude 8, and makes an angle of with
the positive y-axis. Find the component form of v.
a.
b.
c.
d.
e.
689
Chapter 11: Vectors and the Geometry of Space
____
19.
Suppose the guy wire to a 100-foot tower has a tension of 500 pounds. Using the
distances shown in the figure, write the component form of the vector F representing the tension
in the wire. Round numerical values in your answer to the nearest integer.
a.
b.
c.
d.
e.
11.2 Space Coordinates and Vectors in Space
690
11.2 Space Coordinates and Vectors in Space
Answer Section
691 Chapter 11: Vectors and the Geometry of Space
11.3 The Dot Product of Two Vectors
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Given and find .
–8
–4
4
7
–1
____ 2. Given , find .
–6
12
–8
–2
20
____ 3. Given and find .
a.
b.
c.
d.
e.
____ 4. Find the angle between the vectors for u and v given below.
a.
b.
c.
d.
e.
____ 5. Find the angle between the vectors for u and v.
48.22
2.73
87.27
44.97
88.64
11.3 The Dot Product of Two Vectors
692
____ 6.
Determine whether u and v are orthogonal, parallel and pointing in the same
direction, parallel and pointing in opposite directions, or neither.
Orthogonal
Parallel, opposite direction
Parallel, same direction
Neither parallel nor orthogonal
____ 7. Determine whether u and v are orthogonal, parallel or neither.
Parallel, same direction
Orthogonal
Neither parallel nor orthogonal
Parallel, opposite direction
____ 8. Find the direction cosines of the vector u given below.
a.
b.
c.
d.
e.
