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48 CHAPTER 3 VECTORS

Calculations:To evaluate Eqs. 3-16 and 3-17, we find the xand

ycomponents of each displacement. As an example, the com-

ponents for the first displacement are shown in Fig. 3-16c.We

draw similar diagrams for the other two displacements and

then we apply the xpart of Eq. 3-5 to each displacement, using

angles relative to the positive direction of the xaxis:

dlx(6.00 m) cos 40°4.60 m

d 2 x(8.00 m) cos (60°)4.00 m

d 3 x(5.00 m) cos 0°5.00 m.

Equation 3-16 then gives us

dnet,x4.60 m 4.00 m 5.00 m

13.60 m.

Similarly, to evaluate Eq. 3-17, we apply the ypart of Eq. 3-5

to each displacement:

dly(6.00 m) sin 40° = 3.86 m

d 2 y(8.00 m) sin (60°) = 6.93 m

d 3 y(5.00 m) sin 0°0 m.

Equation 3-17 then gives us

dnet,y3.86 m 6.93 m 0 m

3.07 m.

Next we use these components of netto construct the vec-

tor as shown in Fig. 3-16d: the components are in a head-to-

tail arrangement and form the legs of a right triangle, and

d

:

Sample Problem 3.03 Searching through a hedge maze

A hedge maze is a maze formed by tall rows of hedge.
After entering, you search for the center point and then
for the exit. Figure 3-16ashows the entrance to such a
maze and the first two choices we make at the junctions
we encounter in moving from point ito point c. We un-
dergo three displacements as indicated in the overhead
view of Fig. 3-16b:

d 1 6.00 m  1 40°

d 2 8.00 m  2 30°

d 3 5.00 m  3 0°,

where the last segment is parallel to the superimposed
xaxis. When we reach point c, what are the magnitude and
angle of our net displacement netfrom point i?

KEY IDEAS

(1) To find the net displacement net, we need to sum the
three individual displacement vectors:

net 1  2  3.

(2) To do this, we first evaluate this sum for the xcompo-
nents alone,
dnet,xdlxd 2 xd 3 x, (3-16)

and then the ycomponents alone,

dnet,yd 1 yd 2 yd 3 y. (3-17)

d

:

d

:

d

:

d

:

d

:

d

:

Figure 3-16(a) Three displacements through a hedge maze. (b) The displacement vectors. (c) The first displacement vector and its
components. (d) The net displacement vector and its components.

(a)

y

x

d 1 y

d 1 x

(c)

a

b

c

i (b)

y

x

a

b

c

i

u 1

u 2

y

x

dnet,x

dnet,y

c

(d)

d (^1) d
2
d 3
d 1
i
dnet
Three
vectors
First
vector
Net
vector