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Physics For Scientists And Engineers 6E - part 18

Finally, displacement

c, whose magnitude is 195 km, has the

components

c cos(180$) " (195 km)(# 1) " # 195 km

c sin(180$) " 0

Therefore, the components of the position vector

R from

the starting point to city C are

152 km # 52.3 km # 195 km

87.5 km % 144 km % 0 " 232 km

95.3 km

Solution The resultant displacement for the trip

R " A % B

has components given by Equation 3.15:

In unit–vector form, we can write the total displacement as

Using Equations 3.16 and 3.17, we find that the vector

has a magnitude of 41.3 km and is directed 24.1° north of
east.

Let us finalize. The units of

R are km, which is reason-

able for a displacement. Looking at the graphical represen-
tation in Figure 3.19, we estimate that the final position of
the hiker is at about (38 km, 17 km) which is consistent with
the components of

R in our final result. Also, both compo-

nents of

R are positive, putting the final position in the first

quadrant of the coordinate system, which is also consistent
with Figure 3.19.

(37.7ˆ

i % 16.9ˆj) km

R "

16.9 km

" #

17.7 km % 34.6 km "

37.7 km

17.7 km % 20.0 km "

SECTION 3.4 • Components of a Vector and Unit Vectors

Figure 3.19 (Example 3.5) The total displace-

ment of the hiker is the vector R " A % B.

y(km)

x(km)

60.0

45.0

° 20

Tower

Car

–10

–20

Tent

Example 3.6 Let’s Fly Away!

100 150 200

y(km)

150

250

200

100

110

20.0

30.0

x(km)

(B)

Determine the components of the hiker’s resultant dis-

placement

R for the trip. Find an expression for R in terms

of unit vectors.

34.6 km

B sin 60.0$ " (40.0 km)(0.866) "

20.0 km

B cos 60.0$ " (40.0 km)(0.500) "

A commuter airplane takes the route shown in Figure 3.20.
First, it flies from the origin of the coordinate system shown
to city A, located 175 km in a direction 30.0° north of east.
Next, it flies 153 km 20.0° west of north to city B. Finally, it
flies 195 km due west to city C. Find the location of city C
relative to the origin.

Solution Once again, a drawing such as Figure 3.20 allows us
to conceptualize the problem. It is convenient to choose the co-
ordinate system shown in Figure 3.20, where the x axis points
to the east and the y axis points to the north. Let us denote the
three consecutive displacements by the vectors

a, b, and c.

We can now categorize this problem as being similar to

Example 3.5 that we have already solved. There are two pri-
mary differences. First, we are adding three vectors instead
of two. Second, Example 3.5 guided us by first asking for the
components in part (A). The current Example has no such
guidance and simply asks for a result. We need to analyze the
situation and choose a path. We will follow the same pattern
that we did in Example 3.5, beginning with finding the com-
ponents of the three vectors

a, b, and c. Displacement a has

a magnitude of 175 km and the components

Displacement

b, whose magnitude is 153 km, has the com-

ponents

b sin(110$) " (153 km)(0.940) " 144 km

b cos(110$) " (153 km)(#0.342) " #52.3 km

a sin(30.0$) " (175 km)(0.500) " 87.5 km

a cos(30.0$) " (175 km)(0.866) " 152 km

Figure 3.20 (Example 3.6) The airplane starts

at the origin, flies first to city A, then to city B,

and finally to city C.

Investigate this situation at the Interactive Worked Example link at http://www.pse6.com.

CHAPTE R 3 • Vectors

Scalar quantities are those that have only a numerical value and no associated direc-
tion. Vector quantities have both magnitude and direction and obey the laws of vector
addition. The magnitude of a vector is always a positive number.

When two or more vectors are added together, all of them must have the same

units and all of them must be the same type of quantity. We can add two vectors A and
B graphically. In this method (Fig. 3.6), the resultant vector R " A % B runs from the
tail of A to the tip of B.

A second method of adding vectors involves components of the vectors. The x com-

ponent A

of the vector A is equal to the projection of A along the x axis of a coordi-

nate system, as shown in Figure 3.13, where A

A cos !. The y component A

of A is

the projection of A along the y axis, where A

A sin !. Be sure you can determine

which trigonometric functions you should use in all situations, especially when ! is de-
fined as something other than the counterclockwise angle from the positive x axis.

If a vector A has an x component A

and a y component A

, the vector can be ex-

pressed in unit–vector form as A " A

ˆi % A

ˆj. In this notation, ˆi is a unit vector point-

ing in the positive x direction, and ˆj is a unit vector pointing in the positive y direction.
Because ˆi and ˆj are unit vectors,

!ˆi! = !ˆj! = 1.

We can find the resultant of two or more vectors by resolving all vectors into their x

and y components, adding their resultant x and y components, and then using the
Pythagorean theorem to find the magnitude of the resultant vector. We can find the
angle that the resultant vector makes with respect to the x axis by using a suitable
trigonometric function.

S U M M A R Y

1. Two vectors have unequal magnitudes. Can their sum be

zero? Explain.

2. Can the magnitude of a particle’s displacement be greater

than the distance traveled? Explain.

3. The magnitudes of two vectors A and B are A " 5 units

and B " 2 units. Find the largest and smallest values possi-
ble for the magnitude of the resultant vector R " A % B .

4. Which of the following are vectors and which are not:

force, temperature, the volume of water in a can, the rat-
ings of a TV show, the height of a building, the velocity of
a sports car, the age of the Universe?

A vector A lies in the xy plane. For what orientations of A
will both of its components be negative? For what orienta-
tions will its components have opposite signs?

Q U E S T I O N S

In unit–vector notation,

R " (# 95.3ˆi % 232ˆj) km . Using

Equations 3.16 and 3.17, we find that the vector

R has a

magnitude of 251 km and is directed 22.3° west of north.

To finalize the problem, note that the airplane can reach

city C from the starting point by first traveling 95.3 km due
west and then by traveling 232 km due north. Or it could
follow a straight-line path to C by flying a distance R "
251 km in a direction 22.3° west of north.

What If?

After landing in city C, the pilot wishes to return to

the origin along a single straight line. What are the compo-
nents of the vector representing this displacement? What
should the heading of the plane be?

Answer The desired vector

H (for Home!) is simply the

negative of vector

H " # R " (% 95.3ˆi # 232ˆj) km

The heading is found by calculating the angle that the vec-
tor makes with the x axis:

This gives a heading angle of ! " # 67.7°, or 67.7° south of
east.

tan ! "

232 m

95.3 m

" #

2.43

Take a practice test for

this chapter by clicking on
the Practice Test link at
http://www.pse6.com.

Problems

6. A book is moved once around the perimeter of a tabletop

with the dimensions 1.0 m ) 2.0 m. If the book ends up
at its initial position, what is its displacement? What is the
distance traveled?

7. While traveling along a straight interstate highway you no-

tice that the mile marker reads 260. You travel until you
reach mile marker 150 and then retrace your path to the
mile marker 175. What is the magnitude of your resultant
displacement from mile marker 260?

8. If the component of vector A along the direction of vector

B is zero, what can you conclude about the two vectors?

9. Can the magnitude of a vector have a negative value?

Explain.

10. Under what circumstances would a nonzero vector lying

in the xy plane have components that are equal in magni-
tude?

11. If A " B, what can you conclude about the components of

A and B?

Is it possible to add a vector quantity to a scalar quantity?
Explain.

13. The resolution of vectors into components is equivalent to

replacing the original vector with the sum of two vectors,
whose sum is the same as the original vector. There are an
infinite number of pairs of vectors that will satisfy this con-
dition; we choose that pair with one vector parallel to the
x axis and the second parallel to the y axis. What difficul-
ties would be introduced by defining components relative
to axes that are not perpendicular—for example, the
x axis and a y axis oriented at 45° to the x axis?

14. In what circumstance is the x component of a vector given

by the magnitude of the vector times the sine of its direc-
tion angle?

12.

= straightforward, intermediate, challenging

= full solution available in the Student Solutions Manual and Study Guide

= coached solution with hints available at http://www.pse6.com

= computer useful in solving problem

= paired numerical and symbolic problems

P R O B L E M S

Section 3.1 Coordinate Systems

The polar coordinates of a point are r " 5.50 m and

! "

240°. What are the Cartesian coordinates of this point?

2. Two points in a plane have polar coordinates (2.50 m,

30.0°) and (3.80 m, 120.0°). Determine (a) the Cartesian
coordinates of these points and (b) the distance between
them.

A fly lands on one wall of a room. The lower left-hand cor-
ner of the wall is selected as the origin of a two-dimen-
sional Cartesian coordinate system. If the fly is located at
the point having coordinates (2.00, 1.00) m, (a) how far is
it from the corner of the room? (b) What is its location in
polar coordinates?

4. Two points in the xy plane have Cartesian coordinates

(2.00, # 4.00) m and (# 3.00, 3.00) m. Determine (a) the
distance between these points and (b) their polar coordi-
nates.

5. If the rectangular coordinates of a point are given by (2, y)

and its polar coordinates are (r, 30°), determine y and r.

If the polar coordinates of the point (x, y) are (r, !), deter-

mine the polar coordinates for the points: (a) (# x, y ),
(b) (# 2x, #2y), and (c) (3x, # 3y).

Section 3.2 Vector and Scalar Quantities
Section 3.3 Some Properties of Vectors

A surveyor measures the distance across a straight river by
the following method: starting directly across from a tree

on the opposite bank, she walks 100 m along the river-
bank to establish a baseline. Then she sights across to the
tree. The angle from her baseline to the tree is 35.0°. How
wide is the river?

8. A pedestrian moves 6.00 km east and then 13.0 km north.

Find the magnitude and direction of the resultant dis-
placement vector using the graphical method.

9. A plane flies from base camp to lake A, 280 km away, in a

direction  of  20.0° north  of  east.  After  dropping  off  sup-
plies  it  flies  to  lake  B,  which  is  190 km  at  30.0° west  of
north  from  lake  A.  Graphically  determine  the  distance
and direction from lake B to the base camp.

10. Vector A has a magnitude of 8.00 units and makes an an-

gle of 45.0° with the positive x axis. Vector B also has a
magnitude of 8.00 units and is directed along the negative
x axis. Using graphical methods, find (a) the vector sum
A % B and (b) the vector difference A # B.

A skater glides along a circular path of radius 5.00 m.

If he coasts around one half of the circle, find (a) the
magnitude of the displacement vector and (b) how far the
person skated. (c) What is the magnitude of the displace-
ment if he skates all the way around the circle?

12. A force F

of magnitude 6.00 units acts at the origin in a

direction 30.0° above the positive x axis. A second force F

of magnitude 5.00 units acts at the origin in the direction
of the positive y axis. Find graphically the magnitude and
direction of the resultant force F

13.

Arbitrarily define the “instantaneous vector height” of a

person as the displacement vector from the point halfway

11.

CHAPTE R 3 • Vectors

3.00 m

30.0

Figure P3.15 Problems 15 and 37.

Figure P3.18

100 m

30.0

between his or her feet to the top of the head. Make an or-
der-of-magnitude estimate of the total vector height of all
the people in a city of population 100 000 (a) at 10 o’clock
on a Tuesday morning, and (b) at 5 o’clock on a Saturday
morning. Explain your reasoning.

14. A dog searching for a bone walks 3.50 m south, then runs

8.20 m at an angle 30.0° north of east, and finally walks
15.0 m west. Find the dog’s resultant displacement vector
using graphical techniques.

Each of the displacement vectors A and B shown in

Fig. P3.15 has a magnitude of 3.00 m. Find graphically (a)
A % B, (b) A # B, (c) B # A, (d) A # 2B. Report all angles
counterclockwise from the positive x axis.

15.

A vector has an x component of # 25.0 units and a y

component of 40.0 units. Find the magnitude and direc-
tion of this vector.

20. A person walks 25.0° north of east for 3.10 km. How far

would she have to walk due north and due east to arrive at
the same location?

Obtain expressions in component form for the position
vectors having the following polar coordinates: (a) 12.8 m,
150° (b) 3.30 cm, 60.0° (c) 22.0 in., 215°.

22. A displacement vector lying in the xy plane has a magni-

tude of 50.0 m and is directed at an angle of 120° to the
positive x axis. What are the rectangular components of
this vector?

23. A girl delivering newspapers covers her route by traveling

3.00 blocks west, 4.00 blocks north, and then 6.00 blocks
east. (a) What is her resultant displacement? (b) What is
the total distance she travels?

24.

In 1992, Akira Matsushima, from Japan, rode a unicycle

across  the  United  States,  covering  about  4  800 km  in  six
weeks.  Suppose  that,  during  that  trip,  he  had  to  find  his
way  through  a  city  with  plenty  of  one-way  streets.  In  the
city  center,  Matsushima  had  to  travel  in  sequence  280 m
north, 220 m east, 360 m north, 300 m west, 120 m south,
60.0 m east, 40.0 m south, 90.0 m west (road construction)
and then 70.0 m north. At that point, he stopped to rest.
Meanwhile,  a  curious  crow  decided  to  fly  the  distance
from his starting point to the rest location directly (“as the
crow flies”). It took the crow 40.0 s to cover that distance.
Assuming  the  velocity  of  the  crow  was  constant,  find  its
magnitude and direction.

25. While exploring a cave, a spelunker starts at the entrance

and moves the following distances. She goes 75.0 m north,
250 m east, 125 m at an angle 30.0° north of east, and
150 m south. Find the resultant displacement from the
cave entrance.

26. A map suggests that Atlanta is 730 miles in a direction of

5.00° north of east from Dallas. The same map shows that
Chicago is 560 miles in a direction of 21.0° west of north
from Atlanta. Modeling the Earth as flat, use this informa-
tion to find the displacement from Dallas to Chicago.

27. Given the vectors A " 2.00ˆi % 6.00ˆj and B " 3.00ˆi #

2.00ˆj, (a) draw the vector sum C " A % B and the vector
difference D " A # B. (b) Calculate C and D, first in
terms of unit vectors and then in terms of polar coordi-
nates, with angles measured with respect to the % x axis.

28. Find the magnitude and direction of the resultant of three

displacements having rectangular components (3.00,
2.00) m, (# 5.00, 3.00) m, and (6.00, 1.00) m.

A man pushing a mop across a floor causes it to undergo
two displacements. The first has a magnitude of 150 cm
and makes an angle of 120° with the positive x axis. The re-
sultant displacement has a magnitude of 140 cm and is di-
rected at an angle of 35.0° to the positive x axis. Find the
magnitude and direction of the second displacement.

30. Vector A has x and y components of # 8.70 cm and 15.0

cm, respectively; vector B has x and y components of 13.2
cm and # 6.60 cm, respectively. If A # B % 3C " 0, what
are the components of C?

29.

21.

19.

16. Three displacements are A " 200 m, due south; B "

250 m,  due  west;  C " 150 m,  30.0° east  of  north.  Con-
struct  a  separate  diagram  for  each  of  the  following  possi-
ble  ways  of  adding  these  vectors:  R

A % B % C; R

B % C % A; R

C % B % A.

A  roller  coaster  car  moves  200  ft  horizontally,  and  then
rises  135  ft  at  an  angle  of  30.0° above  the  horizontal.  It
then travels 135 ft at an angle of 40.0° downward. What is
its  displacement  from  its  starting  point?  Use  graphical
techniques.

Section 3.4 Components of a Vector and Unit Vectors

18. Find the horizontal and vertical components of the 100-m

displacement of a superhero who flies from the top of a
tall building following the path shown in Fig. P3.18.

17.

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