Physics (Part 1 - Part 3)

3.2 | Components of a Vector 61
So x-components are added only to x-components, and y-components only to
y-components. The magnitude and direction of R S can subsequently be found with
Equations 3.3 and 3.4.
Subtracting two vectors works the same way because it’s a matter of adding the
negative of one vector to another vector. You should make a rough sketch when
adding or subtracting vectors, in order to get an approximate geometric solution
as a check.
■ EXAMPLE 3.3
Take a Hike
GOAL Add vectors algebraically and find the
resultant vector.
PROBLEM A hiker begins a trip by first walking
25.0 km 45.0° south of east from her base camp.
On the second day she walks 40.0 km in a direction
60.0° north of east, at which point she discovers
a forest ranger’s tower. (a) Determine the
components of the hiker’s displacements in the
first and second days. (b) Determine the components
of the hiker’s total displacement for the
trip. (c) Find the magnitude and direction of the
displacement from base camp.
STRATEGY This problem is just an application
of vector addition using components, Equations
3.5. We denote the displacement vectors on the
first and second days by A S
and B S , respectively.
Using the camp as the origin of the coordinates, we get the vectors shown in Figure 3.12a. After finding x- and y-components
for each vector, we add them “componentwise.” Finally, we determine the magnitude and direction of the resultant
vector R S , using the Pythagorean theorem and the inverse tangent function.
SOLUTION
(a) Find the components of A S .
Camp 0
y (km)
20
10
10
20
a
S
A
W
R S
N
S
E
45.0 20 30 40
60.0
B S
Tower
x (km)
y (km)
20
10
R y = 16.9 km
0
u
x (km)
O 10 20 30 40
R
10 x = 37.7 km
20
Figure 3.12 (Example 3.3) (a) Hiker’s path and the resultant vector. (b) Components
of the hiker’s total displacement from camp.
b
R S
Use Equations 3.2 to find the components of A S :
Find the components of B S :
(b) Find the components of the resultant vector,
R S 5 A S 1 B S .
To find R x
, add the x-components of A S and B S :
To find R y
, add the y-components of A S and B S :
(c) Find the magnitude and direction of R S .
Use the Pythagorean theorem to get the magnitude:
Calculate the direction of R S using the inverse tangent
function:
A x
5 A cos (245.0°) 5 (25.0 km)(0.707) 5 17.7 km
A y
5 A sin (245.0°) 5 2(25.0 km)(0.707) 5 217.7 km
B x
5 B cos 60.0° 5 (40.0 km)(0.500) 5 20.0 km
B y
5 B sin 60.0° 5 (40.0 km)(0.866) 5 34.6 km
R x
5 A x
1 B x
5 17.7 km 1 20.0 km 5 37.7 km
R y
5 A y
1 B y
5 217.7 km 1 34.6 km 5 16.9 km
R 5 "R x 2 1 R y 2 5 "137.7 km2 2 1 116.9 km2 2 5 41.3 km
u5tan 21 16.9 km
a
37.7 km b 5 24.1° (Continued)
REMARKS Figure 3.12b shows a sketch of the components of R S and their directions in space. The magnitude and direction
of the resultant can also be determined from such a sketch.
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