Download SECTION 7-3 Geometric Vectors

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7 Additional Topics in Trigonometry
miles above the surface of the Earth (see figure). (A satellite
in geostationary orbit remains stationary above a fixed
point on the surface of the Earth.) Radio signals are sent
from the tracking station by way of the satellites to the shuttle, and vice versa. This system allows the tracking station
to be in contact with the shuttle over most of the Earth’s
surface. How far to the nearest 100 miles is one of the geostationary satellites from the White Sands tracking station
W? The radius of the Earth is 3,964 miles.
S
★
48. Space Science. A satellite S, in circular orbit around the
Earth, is sighted by a tracking station T (see figure). The
distance TS is determined by radar to be 1,034 miles, and
the angle of elevation above the horizon is 32.4°. How high
is the satellite above the Earth at the time of the sighting?
The radius of the Earth is 3,964 miles.
S
T
Horizon
S
R
W
Earth
C
C
SECTION
7-3
Geometric Vectors
•
•
•
•
Geometric Vectors and Vector Addition
Velocity Vectors
Force Vectors
Resolution of Vectors into Vector Components
Many physical quantities, such as length, area, or volume, can be completely specified by a single real number. Other quantities, such as directed distances, velocities,
and forces, require for their complete specification both a magnitude and a direction. The former are often called scalar quantities, and the latter are called vector
quantities.
In this section we limit our discussion to the intuitive idea of geometric vectors
in a plane. In Section 7-4 we introduce algebraic vectors, a first step in the generalization of a concept that has far-reaching consequences. Vectors are widely used in
many areas of science and engineering.
• Geometric Vectors
and Vector Addition
P
v
O
, or v.
FIGURE 1 Vector OP
A line segment to which a direction has been assigned is called a directed line segment. A geometric vector is a directed line segment and is represented by an arrow
(see Fig. 1). A vector with an initial point O and a terminal point P (the end with
. Vectors are also denoted by a boldface letter, such
the arrowhead) is denoted by OP
as v. Since it is difficult to write boldface on paper, we suggest that you use an arrow
over a single letter, such as v , when you want the letter to denote a vector.
, denoted by OP
, v
or v, is the length of the
The magnitude of the vector OP
directed line segment. Two vectors have the same direction if they are parallel and
point in the same direction. Two vectors have opposite direction if they are parallel
and point in opposite directions. The zero vector, denoted by 0 or 0, has a magni-
7-3
u
v
v
u
FIGURE 2 Vector addition: tail-totip rule.
v
u
v
u
FIGURE 3 Vector addition: parallelogram rule.
EXPLORE-DISCUSS 1
Geometric Vectors
525
tude of zero and an arbitrary direction. Two vectors are equal if they have the same
magnitude and direction. Thus, a vector may be translated from one location to
another as long as the magnitude and direction do not change.
The sum of two vectors u and v can be defined using the tail-to-tip rule: Translate v so that its tail end (initial point) is at the tip end (terminal point) of u. Then,
the vector from the tail end of u to the tip end of v is the sum, denoted by u v,
of the vectors u and v (see Fig. 2).
The sum of two nonparallel vectors also can be defined using the parallelogram
rule: The sum of two nonparallel vectors u and v is the diagonal of the parallelogram formed using u and v as adjacent sides (see Fig. 3). If u and v are parallel, use
the tail-to-tip rule.
Both rules give the same sum. The choice of which rule to use depends on the
situation and what seems most natural.
The vector u v is also called the resultant of the two vectors u and v, and u
and v are called vector components of u v. It is useful to observe that vector addition is commutative and associative. That is, u v v u and u (v w) (u v) w.
If a, b and c represent three arbitrary geometric vectors, illustrate using either definition of vector addition that:
1. a b b a
2. a (b c) (a b) c
• Velocity Vectors
A vector that represents the direction and speed of an object in motion is called a
velocity vector. Problems involving objects in motion often can be analyzed using
vector methods. Many of these problems involve the use of a navigational compass,
which is marked clockwise in degrees starting at north as indicated in Figure 4.
FIGURE 4
N, 0
W, 270
90, E
S, 180
EXAMPLE 1
Apparent and Actual Velocity
An airplane has a compass heading (the direction the plane is pointing) of 85° and
an airspeed (relative to the air) of 140 miles per hour. The wind is blowing from north
to south at 66 miles per hour. The velocity of a plane relative to the air is called
apparent velocity, and the velocity relative to the ground is called resultant, or
actual velocity. The resultant velocity is the vector sum of the apparent velocity and
the wind velocity. Find the resultant velocity; that is, find the actual speed and
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7 Additional Topics in Trigonometry
direction of the airplane relative to the ground. Directions are given to the nearest
degree and magnitudes to 2 significant digits.
Solution
Geometric vectors [Fig. 5 (a)] are used to represent the apparent velocity vector and
the wind velocity vector. Add the two vectors using the tail-to-tip method of addition
of vectors to obtain the resultant (actual) velocity vector [Fig. 5 (b)]. From the vector diagram [Fig. 5 (b)], we obtain the triangle in Figure 6 and solve for , c, and .
FIGURE 5
N
N
Apparent
velocity
85
Wind
velocity
Apparent
velocity
85
180
Actual
velocity
(a)
FIGURE 6
Wind
velocity
(b)
140
66
c
Solve for Since the wind velocity vector is parallel to the north–south line, 85° [alternate
interior angles of two parallel lines cut by a transversal are equal—see Fig. 5 (b)].
Solve for c
Use the law of cosines:
c2 a2 b2 2ab cos c a2 b2 2ab cos 662 1402 2(66)(140) cos 85°
150 miles per hour
Solve for Speed relative to the ground.
Use the law of sines:
sin sin a
c
a sinb sin 85
66 150
26°
sin1
sin1
Actual heading 85° 85° 26° 111°
Thus, the magnitude and direction of the resultant velocity vector are 150 miles per
hour and 111°, respectively. That is, the plane, relative to the ground, is traveling at
150 miles per hour in a direction of 111°.
7-3
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Geometric Vectors
Matched Problem 1
A river is flowing southwest (225°) at 3.0 miles per hour. A boat crosses the river
with a compass heading of 90°. If the speedometer on the boat reads 5.0 miles per
hour (the boat’s speed relative to the water), what is the resultant velocity? That is,
what is the boat’s actual speed and direction relative to the ground? Directions are to
the nearest degree, and magnitudes are to 2 significant digits.
• Force Vectors
A vector that represents the direction and magnitude of an applied force is called a
force vector. If an object is subjected to two forces, then the sum of these two forces,
the resultant force, is a single force. If the resultant force replaced the original two
forces, it would act on the object in the same way as the two original forces taken
together. In physics it is shown that the resultant force vector can be obtained using
vector addition to add the two individual force vectors. It seems natural to use the
parallelogram rule for adding force vectors, as is illustrated in the next example.
EXAMPLE 2
Finding the Resultant Force
Two forces of 30 and 70 pounds act on a point in a plane. If the angle between the
force vectors is 40°, what are the magnitude and direction (relative to the 70-pound
force) of the resultant force? The magnitudes of the forces are to 2 significant digits
and the angles to the nearest degree.
Solution
We start with a diagram (Fig. 7), letting geometric vectors represent the various forces:
FIGURE 7
A
B
R
30 pounds
40
O
70 pounds
C
Because adjacent angles in a parallelogram are supplementary, the measure of angle
OCB 180° 40° 140°. We can now find the magnitude of the resultant vector
R using the law of cosines (see Fig. 8):
R R2 302 702 2(30)(70) cos 140°
95 pounds
140
R 302 702 2(30)(70) cos 140°
70
FIGURE 8
To find , the direction of R, we use the law of sines (see Fig. 9):
sin sin 140°
30
95
sin 30 sin 140°
95
95
140
70
FIGURE 9
30
30
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7 Additional Topics in Trigonometry
sin1
30 sin95140° 12°
Thus, the two given forces are equivalent to a single force of 95 pounds in the direction of 12° (relative to the 70-pound force).
Matched Problem 2
• Resolution of
Vectors into Vector
Components
EXAMPLE 3
Repeat Example 2 using an angle of 100° between the two forces.
Instead of adding vectors, many problems require the resolution of vectors into components. As we indicated earlier, whenever a vector is expressed as the sum or resultant of two vectors, the two vectors are called vector components of the given vector. Example 3 illustrates an application of the process of resolving a vector into vector
components.
Resolving a Force Vector into Components
A car weighing 3,210 pounds is on a driveway inclined 20.2° to the horizontal.
Neglecting friction, find the magnitude of the force parallel to the driveway that will
keep the car from rolling down the hill.
Solution
We start by drawing a vector diagram (Fig. 10):
FIGURE 10
y
ewa
Driv
A
D
20.2
C
B
3,210 pounds
acts in a downward direction and represents the weight of
The force vector DB
DA
, where DC
is the perpendicular component of DB
the car. Note that DB DC
relative to the driveway and DA is the parallel component of DB relative to the
driveway.
To keep the car at D from rolling down the hill, we need a force with the mag but oppositely directed. To find DA
, we first observe that ABD nitude of DA
20.2°. This is true because ABD and the driveway angle have the same complement, ADB.
7-3
sin 20.2° Geometric Vectors
529
DA
3,210
3,210 sin 20.2°
DA
1,110 pounds
in Example 3.
Find the magnitude of the perpendicular component of DB
Matched Problem 3
Answers to Matched Problems
1. Resultant velocity: magnitude 3.6 mph, direction 126°
2. R 71 lb, 25°
3,010 lb
3. DC
EXERCISE
7-3
7. u v 390 miles per hour, 6°
Express all angle measures in decimal degrees.
8. u v 75 miles per hour, 81°
A
Problems 1–8 refer to figures (a) and (b) showing vector
addition for vectors u and v at right angles to each other.
u
v
v
B
Problems 9–16 refer to figures (c) and (d) showing vector
addition for vectors u and v.
u
v
v
u
v
u
v
u
Tail-to-tip rule
Parallelogram rule
u
(a)
(b)
In Problems 1–4, find u v and , given u and v in
figures (a) and (b).
1. u 30 miles per hour, v 72 miles per hour
2. u 216 miles per hour, v 63 miles per hour
u
v
v
u
Tail-to-tip rule
Parallelogram rule
(c)
In Problems 9–12, find u v and , given u, v, and in figures (c) and (d).
9. u 66 grams, v 22 grams, 68°
3. u 29 kilograms, v 29 kilograms
10. u 120 grams, v 84 grams, 44°
4. u 78 kilograms, v 45 kilograms
11. u 21 knots, v 3.2 knots, 53°
In Problems 5–8, find u and v, the magnitudes of the
horizontal and vertical components of u v, given u v
and in figures (a) and (b).
(d)
12. u 8.0 knots, v 2.0 knots, 64°
In Problems 13–16, find u and v, given u v, and in figures (c) and (d).
5. u v 24 pounds, 60°
13. u v 14 kilograms, 25°, 79°
6. u v 48 pounds, 45°
14. u v 33 kilograms, 17°, 43°
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7 Additional Topics in Trigonometry
15. u v 223 miles per hour, 42.3°, 69.4°
river’s current is 2.5 knots, what compass heading should
be maintained while crossing the river? What is the actual
speed of the boat relative to the land?
16. u v 437 miles per hour, 17.8°, 50.5°
★
28. Navigation. An airplane can cruise at 255 miles per hour in
still air. If a steady wind of 46.0 miles per hour is blowing
from the west, what compass heading should the pilot fly in
order for the course of the plane relative to the ground to be
north (0°)? Compute the ground speed for this course.
★
29. Resultant Force. A large ship has gone aground in a harbor
and two tugs, with cables attached, attempt to pull it free. If
one tug pulls with a compass course of 52° and a force of
2,300 pounds and a second tug pulls with a compass course
of 97° and a force of 1,900 pounds, what is the compass direction and the magnitude of the resultant force?
★
30. Resultant Force. Repeat Problem 29 if one tug pulls with
a compass direction of 161° and a force of 2,900 kilograms
and a second tug pulls with a compass direction of 192° and
a force of 3,600 kilograms.
C
In Problems 17–24, determine whether the statement is true
or false. If true, explain why. If false, give a counterexample.
17. The zero vector is perpendicular to every vector.
18. The zero vector is parallel to every vector.
19. Vectors having the same magnitude are equal.
20. Equal vectors have the same magnitude.
21. Equal vectors have the same direction.
22. Perpendicular vectors have the opposite direction.
23. The magnitude of every vector is positive.
24. The magnitude of u v is greater than u.
31. Resolution of Forces. An automobile weighing 4,050
pounds is standing on a driveway inclined 5.5° with the
horizontal.
(A) Find the magnitude of the force parallel to the driveway necessary to keep the car from rolling down the
hill.
(B) Find the magnitude of the force perpendicular to the
driveway.
APPLICATIONS
In navigation problems, refer to the figure of a navigational
compass below:
32. Resolution of Forces. Repeat Problem 31 for a car weighing 2,500 pounds parked on a hill inclined at 15° to the
horizontal.
N, 0
★
W, 270
90, E
33. Resolution of Forces. If two weights are fastened together
and placed on inclined planes as shown in the figure, neglecting friction, which way will they slide?
S, 180
Navigational compass
110 ds
n
pou
In Problems 25–28, assume the north, east, south, and west
directions are exact.
25. Navigation. An airplane is flying with a compass heading
of 285° and an airspeed of 230 miles per hour. A steady
wind of 35 miles per hour is blowing in the direction of
260°. What is the plane’s actual velocity; that is, what is its
speed and direction relative to the ground?
25
★
27. Navigation. Two docks are directly opposite each other on
a southward-flowing river. A boat pilot wishes to go in a
straight line from the east dock to the west dock in a ferryboat with a cruising speed in still water of 8.0 knots. If the
35
34. Resolution of Forces. If two weights are fastened together
and placed on inclined planes as indicated in the figure, neglecting friction, which way will they slide?
26. Navigation. A powerboat crossing a wide river has a compass heading of 25° and speed relative to the water of 15
miles per hour. The river is flowing in the direction of 135°
at 3.9 miles per hour. What is the boat’s actual velocity; that
is, what is its speed and direction relative to the ground?
★
po 85
un
ds
41 ds
un
po
31
po 31
un
ds
41