154
The following example shows the correct handling of the plus and
minus signs, which is usually the main cause of mistakes.
Example: negative components
Question: San Diego is 120 km east and 150 km south of Los
Angeles. An airplane pilot is setting course from San Diego to
Los Angeles. At what angle should she set her course, measured
counterclockwise from east, as shown in the figure.
Solution: If we make the traditional choice of coordinate axes,
with x pointing to the right and y pointing up on the map, then her
.
x is negative, because her final x value is less than her initial x
value. Her
.
y is positive, so we have
.
x=
-
120 km
.
y= 150 km .
If we work by analogy with the previous example, we get
.
=
tan
–1
.
y
.
x
=
tan
–1
–1.25
=
-
51
°
.
According to the usual way of defining angles in trigonometry, a
negative result means an angle that lies clockwise from the x
axis, which would have her heading for the Baja California. What
went wrong. The answer is that when you ask your calculator to
take the arctangent of a number, there are always two valid
possibilities differing by 180
°
. That is, there are two possible
angles whose tangents equal -1.25:
tan 129
°
= -1.25
tan -51
°
= -1.25
You calculator doesn’t know which is the correct one, so it just
picks one. In this case, the one it picked was the wrong one, and
it was up to you to add 180
°
to it to find the right answer.
Discussion Question
In the example above, we dealt with components that were negative. Does it
make sense to talk about positive and negative vectors.
7.3Techniques for Adding Vectors
Addition of vectors given their components
The easiest type of vector addition is when you are in possession of the
components, and want to find the components of their sum.
Example
Question: Given the
.
x and
.
y values from the previous ex-
amples, find the
.
x and
.
y from San Diego to Las Vegas.
Solution:
.
x
total
=
.
x
1
+
.
x
2
= –120 km + 290 km
= 170 km
.
y
total
=
.
y
1
+
.
y
2
= 150 km + 230 km
= 380
Note how the signs of the x components take care of the west-
ward and eastward motions, which partially cancel.
Los
Angeles
Las Vegas
San Diego
Chapter 7Vectors
.
x
.
y
Los
Angeles
|
.
r|
.
San Diego
(negative)