9781118230725.pdf

PROBLEMS 61

67 Let be directed to the east, be directed to the north, and kiˆ jˆ ˆ 72 A fire ant, searching for hot sauce in a picnic area, goes
through three displacements along level ground: lfor 0.40 m
southwest (that is, at 45° from directly south and from directly
west), 2 for 0.50 m due east, 3 for 0.60 m at 60° north of east.
Let the positive xdirection be east and the positive ydirection
be north. What are (a) the xcomponent and (b) the ycompo-
nent of l? Next, what are (c) the xcomponent and (d) the y
component of 2? Also, what are (e) the xcomponent and (f)
theycomponent of 3?
What are (g) the xcomponent, (h) the ycomponent, (i) the
magnitude, and (j) the direction of the ant’s net displacement? If
the ant is to return directly to the starting point, (k) how far and (1)
in what direction should it move?
73 Two vectors are given by 3.05.0 and b2.0iˆ4.0 .ˆj
:
a: iˆ jˆ

d

d :

d :

:

d

:

d

:

d:

a

b

f

Figure 3-38Problem 79.

BOSTON

and Vicinity

Wellesley

Waltham

Brookline

Newton

Arlington

Lexington

Woburn

Medford

Lynn

Salem

Quincy

5 10 km

BOSTON

Massachusetts

Bay

Bank

Walpole

Framingham

Weymouth

Dedham

Winthrop

N

Figure 3-36Problem 68.

be directed upward. What are the values of products (a)  , (b)
(kˆ)(ˆj), and (c) jˆ(ˆj)? What are the directions (such as east

iˆ kˆ

or down) of products (d) ,(e) ()(), and (f) () ()?

68 A bank in downtown Boston is robbed (see the map in
Fig. 3-36). To elude police, the robbers escape by helicopter, mak-
ing three successive flights described by the following displace-
ments: 32 km, 45° south of east; 53 km, 26° north of west; 26 km, 18°
east of south. At the end of the third flight they are captured. In
what town are they apprehended?

kˆ ˆj iˆ ˆj kˆ ˆj

69 A wheel with a radius of 45.0 cm
rolls without slipping along a hori-
zontal floor (Fig. 3-37). At time t 1 ,
the dot Ppainted on the rim of the
wheel is at the point of contact be-
tween the wheel and the floor. At a
later time t 2 , the wheel has rolled
through one-half of a revolution.
What are (a) the magnitude and (b)
the angle (relative to the floor) of
the displacement of P?

70 A woman walks 250 m in the direction 30° east of north, then
175 m directly east. Find (a) the magnitude and (b) the angle of her
final displacement from the starting point. (c) Find the distance she
walks. (d) Which is greater, that distance or the magnitude of her
displacement?

71 A vector has a magnitude 3.0 m and is directed south. What
are (a) the magnitude and (b) the direction of the vector 5.0? What
are (c) the magnitude and (d) the direction of the vector 2.0 ?d:

:d

d

:

P

At time t 1 At time t 2

P

Figure 3-37Problem 69.

Find (a)  , (b) , (c) , and (d) the component of

along the direction of.

74 Vector lies in the yzplane 63.0from the positive direction

of the yaxis, has a positive zcomponent, and has magnitude 3.20

units. Vector lies in the xzplane 48.0from the positive direction

of the xaxis, has a positive zcomponent, and has magnitude 1.40

units. Find (a)  , (b)  , and (c) the angle between and.

75 Find (a) “north cross west,” (b) “down dot south,” (c) “east

cross up,” (d) “west dot west,” and (e) “south cross south.” Let each

“vector” have unit magnitude.

76 A vector , with a magnitude of 8.0 m, is added to a vector ,

which lies along an xaxis. The sum of these two vectors is a third

vector that lies along the yaxis and has a magnitude that is twice

the magnitude of. What is the magnitude of?

77 A man goes for a walk, starting from the origin of an xyz

coordinate system, with the xyplane horizontal and the xaxis east-

ward. Carrying a bad penny, he walks 1300 m east, 2200 m north,

and then drops the penny from a cliff 410 m high. (a) In unit-vector

notation, what is the displacement of the penny from start to its

landing point? (b) When the man returns to the origin, what is the

magnitude of his displacement for the return trip?

78 What is the magnitude of (  ) if a3.90,b2.70,

and the angle between the two vectors is 63.0°?

79 In Fig. 3-38, the magnitude of is 4.3, the magnitude of is

5.4, and 46°. Find the area of the triangle contained between

the two vectors and the thin diagonal line.

b

:

:a

b :a

:

:a

A

:

A

:

A

:

B

:

b

:

b a:

:

b a:

:

:a

b

:

:a

b

:

:a

b

:

(:ab

:

a:b )

:

b

:

a: