419
Addition of Vectors
A
6c
35.
B
66.3˚
cm
m
19 Two links, AC and BC, are pivoted at C, as shown in Fig. 15–30. How far apart are
A and B when angle ACB is 66.3?
20. A robot is used to lift a metal grill from a bin and place it precisely in a housing for
welding. When the robot has placed the grill, a welding robot will make the spot
weld. A concern is raised over the amount of heat reaching the unshielded shoulder joint of the robot. As shown in Fig. 15–31, the robotic arm has a 65.0 cm arm
extended from a base (shoulder) and a second arm 20.0 cm long connected to the
first at the “elbow.” If the angle between the two arms is 115, what is the distance
between the manipulator holding the grill and the shoulder joint?
.8
◆
22
Section 15–5
C
FIGURE 15–30
Geometry
21. Find angles A, B, and C in the quadrilateral in Fig. 15–32.
cm
.0
65
20
22. Find side AB in the quadrilateral in Fig. 15–33.
.0
cm
115˚
23. Two sides of a parallelogram are 22.8 and 37.8 m, and one of the diagonals is 42.7 m.
Find the angles of the parallelogram.
24. Find the lengths of the sides of a parallelogram if its diagonal, which is 125 mm
long, makes angles with the sides of 22.7 and 15.4.
25. Find the lengths of the diagonals of a parallelogram, two of whose sides are 3.75 m FIGURE 15–31
and 1.26 m; their included angle is 68.4.
26. A median of a triangle is a line joining a vertex to the midpoint of the opposite side.
B
cm
175
In triangle ABC, A 62.3, b 112, and the median from C to the midpoint of c is
186. Find c.
A
110
cm
27. The sides of a triangle are 124, 175, and 208. Find the length of the median drawn
to the longest side.
67.2˚
28. The angles of a triangle are in the ratio 3:4:5, and the shortest side is 994. Solve the
255 cm
triangle.
FIGURE 15–32
29. The sides of a triangle are in the ratio 2:3:4. Find the cosine of the largest angle.
30. Two solar panels are to be placed as shown in Fig. 15–34. Find the minimum dis43.0 in.
B
C
tance x so that the first panel will not cast a shadow on the second when the angle
121.0˚
of elevation of the sun is 18.5.
31. Find the overhang x so that the window in Fig. 15–35 will be in complete shade
when the sun is 60 above the horizontal.
Shoulder
163 cm
C
72.1 in.
x
A
85˚
2.5 m
Window
FIGURE 15–33
3.0
0m
41.6˚
41.6˚
x
FIGURE 15–34
Solar panels.
FIGURE 15–35
15–5 Addition of Vectors
In Sec. 7–6 we added two or more nonperpendicular vectors by first resolving each into components, then adding the components, and finally resolving the components into a single resultant.
63.0˚
105 in.
D
420
◆
Chapter 15
Now, by using the law of sines and the law of cosines, we can combine two nonperpendicular vectors directly, with much less time and effort. However, when more than two
vectors must be added, it is faster to resolve each into its x and y components, combine
the x components and the y components, and then find the resultant of those two perpendicular vectors.
In this section we show both methods.
A
47.2˚
Oblique Triangles and Vectors
B
(a)
A
47.2˚
R
B
(b)
B
47.2˚
Vector Diagram
We can illustrate the resultant, or vector sum, of two vectors by means of a diagram.
Suppose that we wish to add vectors A and B in Fig. 15–36(a). If we draw the two vectors tip to tail, as in Fig. 15–36(b), the resultant R will be the vector that will complete
the triangle when drawn from the tail of the first vector to the tip of the second vector.
It does not matter whether vector A or vector B is drawn first; the same resultant will
be obtained either way, as shown in Fig. 15–36(c).
The parallelogram method will give the same result. To add the same two vectors
A and B as before, we first draw the given vectors tail to tail (Fig. 15–37) and complete
a parallelogram by drawing lines from the tips of the given vectors, parallel to the given
vectors. The resultant R is then the diagonal of the parallelogram drawn from the intersections of the tails of the original vectors.
R
A
R
(c)
FIGURE 15–36 Addition of
vectors.
A
47.2˚
B
FIGURE 15–37 Parallelogram method.
Finding the Resultant of Two Nonperpendicular Vectors
Whichever method we use for drawing two vectors, the resultant is one side of an oblique
triangle. To find the length of the resultant and the angle that it makes with one of the original
vectors, we simply solve the oblique triangle by the methods we learned earlier in this chapter.
◆◆◆
Example 27: Two vectors, A and B, make an angle of 47.2 with each other as shown in
Fig. 15–37. If their magnitudes are A 125 and B 146, find the magnitude of the resultant R
and the angle that R makes with vector B.
Solution: We make a vector diagram, either tip to tail or by the parallelogram method.
Either way, we must solve the oblique triangle in Fig. 15–38 for R and . Finding yields
180 47.2 132.8
By the law of cosines,
125
R
R2 (125)2 (146)2 2(125)(146) cos 132.8 61 740
R 248
Then, by the law of sines,
FIGURE 15–38
146
47.2˚
sin sin 132.8
248
125
125 sin 132.8
sin 0.3698
248
21.7
◆◆◆
Section 15–5
◆
421
Addition of Vectors
Addition of Several Vectors
The law of sines and the law of cosines are good for adding two nonperpendicular vectors.
However, when several vectors are to be added, we usually break each into its x and y components and combine them, as in the following example.
◆◆◆
Example 28: Find the resultant of the vectors shown in Fig. 15–39(a).
Solution: The x component of a vector of magnitude V at any angle is
V cos y
and the y component is
V sin 56
.1
42
These equations apply for an angle in any quadrant. We compute and tabulate the x and
y components of each original vector and find the sums of each as shown in the following table.
.0
A
B
148˚
58.0˚
0
232˚
A
B
C
D
R
42.0 cos 58.0 22.3
56.1 cos 148 47.6
52.7 cos 232 32.4
45.3 cos 291 16.2
Rx 41.5
42.0 sin 58.0 35.6
56.1 sin 148 29.7
52.7 sin 232 41.5
45.3 sin 291 42.3
Ry 18.5
291˚
.7
y Component
52
x Component
45.3
Vector
x
C
D
(a)
y
The two vectors Rx and Ry are shown in Fig. 15–39(b). We find their resultant R by the
Pythagorean theorem.
R2 (41.5)2 (18.5)2 2065
R 45.4
We find the angle by
Ry
arctan Rx
18.5
arctan 41.5
Rx = −41.5
0
Ry = −18.5
x
R
(b)
FIGURE 15–39
24.0 or 204
Since our resultant is in the third quadrant, we drop the 24.0 value. Thus the resultant has a
magnitude of 45.4 and a direction of 204. This is often written in the form
R 45.4 204
Exercise 5
◆
◆◆◆
Addition of Vectors
The magnitudes of vectors A and B are given in the following table, as well as the angle between
the vectors. For each, find the magnitude R of the resultant and the angle that the resultant
makes with vector B.
Magnitudes
1.
2.
3.
4.
5.
6.
This is called polar form, which
we’ll cover in Chapter 17.
A
B
Angle
244
1.85
55.9
1.006
4483
35.2
287
2.06
42.3
1.745
5829
23.8
21.8°
136°
55.5°
148.4°
100.0°
146°
In Sec. 7–7 we gave examples
of applications of force vectors,
velocity vectors, and impedance
vectors. You might want to
review that section before trying
the applications here. These
problems are set up in exactly
the same way, only now they will
require the solution of an oblique
triangle rather than a right
triangle.
422
Chapter 15
◆
Oblique Triangles and Vectors
Force Vectors
7. Two forces of 18.6 N and 21.7 N are applied to a point on a body. The angle between the
forces is 44.6. Find the magnitude of the resultant and the angle that it makes with the
larger force.
8. Two forces whose magnitudes are 187 N and 206 N act on an object. The angle between
the forces is 88.4. Find the magnitude of the resultant force.
9. A force of 125 N pulls due west on a body, and a second force pulls N 28.7 W. The resultant
force is 212 N. Find the second force and the direction of the resultant.
10. Forces of 675 N and 828 N act on a body. The smaller force acts due north; the larger force
acts N 52.3 E. Find the direction and the magnitude of the resultant.
11. Two forces of 925 N and 1130 N act on an object. Their lines of action make an angle of
67.2 with each other. Find the magnitude and the direction of their resultant.
12. Two forces of 136 lb. and 251 lb. act on an object with an angle of 53.9 between their lines
of action. Find the magnitude of their resultant and its direction.
13. The resultant of two forces of 1120 N and 2210 N is 2870 N. What angle does the resultant
make with each of the two forces?
14. Three forces are in equilibrium: 212 N, 325 N, and 408 N. Find the angles between their
lines of action.
Velocity Vectors
15. As an airplane heads west with an air speed of 325 km/h; a wind with a speed of 35.0 km/h
causes the plane to travel slightly south of west with a ground speed of 305 km/h. In what
direction is the wind blowing? In what direction does the plane travel?
See Fig. 7–45 for definitions of
the flight terms used in these
problems.
16 A boat heads S 15.0 E on a river that flows due west. The boat travels S 11.0 W with
a speed of 25.0 km/h. Find the speed of the current and the speed of the boat in still
water.
17. A pilot wishes to fly in the direction N 45.0 E. The wind is from the west at 36.0 km/h, and
the plane’s speed in still air is 388 km/h. Find the heading and the ground speed.
18. The heading of a plane is N 27.7 E, and its air speed is 255 mi./h. If the wind is blowing
from the south with a velocity of 42.0 mi./h, find the actual direction of travel of the plane
and its ground speed.
19. A plane flies with a heading of N 48.0 W and an air speed of 584 km/h. It is driven
from its course by a wind of 58.0 km/h from S 12.0 E. Find the ground speed and
the drift angle of the plane.
I
I2
Current and Voltage Vectors
132˚
I1
38
7.
12.5
15.6˚
FIGURE 15–40
I
Z1
I1
Z2
I2
20. We will see later that it is possible to represent an alternating current or voltage by
a vector whose length is equal to the maximum amplitude of the current or voltage,
placed at an angle that we later define as the phase angle. Then to add two alternating
currents or voltages, we add the vectors representing those voltages or currents in the
same way that we add force or velocity vectors.
A current I1 is represented by a vector of magnitude 12.5 A at an angle of 15.6,
and a second current I2 is represented by a vector of magnitude 7.38 A at an angle
of 132, as shown in Fig. 15–40. Find the magnitude and the direction of the sum of
these currents, represented by the vector I.
21. Figure 15–41 shows two impedances in parallel, with the currents in each represented by
I1 18.4 A
at 51.5
I2 11.3 A
at 0
and
FIGURE 15–41
423
Review Problems
The current I will be the vector sum of I1 and I2. Find the magnitude and the direction of
the vector representing I.
22. Figure 15–42 shows two impedances in series, with the voltage drop V1 equal to 92.4 V at
71.5 and V2 equal to 44.2 V at 53.8. Find the magnitude and the direction of the vector
representing the total drop V.
Find the resultant of each of the following sets of vectors.
V
V2
23. 273 34.0, 179 143 , 203 225 , 138 314
24. 72.5 284 , 28.5 331 , 88.2 104 , 38.9 146
FIGURE 15–42
Case Study Discussion—Bracket for Solar Panel Frame
The bracket represents a SAS triangle. The minimum length of the opposite side
can be calculated using the lower angle and the formula c2 a2 b2 2ab cos C,
where a and b are the known sides, 35 cm and 20 cm, and c is the side opposite the
angle formed at the junction of sides a and b, angle C. The lower value of c would be
c 兹 a2 b2 2ab cos C
兹 352 202 2(35)(20) cos 35 21.9 cm
The longest side can be calculated using the wider angle of 55
c 兹 a2 b2 2ab cos C
兹 352 202 2(35)(20) cos 55 28.7 cm
So the side opposite this changing angle can range from 21.9 cm to 28.7 cm.
CHAPTER 15 REVIEW PROBLEMS
◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆
Solve oblique triangle ABC if:
C 135
A 92.4
B 38.4
B 22.6
a 44.9
a 129
a 1.84
a 2840
b 39.1
c 83.6
c 2.06
b 1170
5. A 132
b 38.2
c 51.8
1.
2.
3.
4.
Z1
Z2
Addition of Several Vectors
◆◆◆
V1
In what quadrant(s) will the terminal side of lie if:
6. 227
7. 45
8. 126
10. tan is negative
Without using book or calculator, state the algebraic sign of:
11. tan 275
12. sec(58)
13. cos 183
14. cos 45
15. sin 300
9. 170
Write the sin, cos, and tan, to three significant digits, for the angle whose terminal side passes
through the given point.
16. (2, 5)
17. (3, 4)
18. (5, 1)
Two vectors of magnitudes A and B are separated by an angle . Find the resultant and the angle
that the resultant makes with vector B.
19. A 837
B 527
58.2
20. A 2.58
B 4.82
82.7
21. A 44.9
B 29.4
155