153
Example: finding the magnitude and angle from the components
Question: Given that the
.
r vector from LA to Las Vegas has
.
x=290 km and
.
y=230 km, how would we find the magnitude
and direction of
.
r.
Solution: We find the magnitude of
.
r from the Pythagorean
theorem:
|
.
r|=
.
x
2
+
.
y
2
= 370 km
We know all three sides of the triangle, so the angle
.
can be
found using any of the inverse trig functions. For example, we
know the opposite and adjacent sides, so
.
=
tan
–1
.
y
.
x
= 38
°
.
Example: finding the components from the magnitude and angle
Question: Given that the straight-line distance from Los Angeles
to Las Vegas is 370 km, and that the angle
.
in the figure is 38
°
,
how can the x and y components of the
.
r vector be found.
Solution: The sine and cosine of
.
relate the given information to
the information we wish to find:
cos
.
=
.
x
.
r
sin
.
=
.
y
.
r
Solving for the unknowns gives
.
x=
.
r
cos
.
= 290 km
.
y=
.
r
sin
.
= 230 km
Los
Angeles
Las Vegas
.
x
.
y
|
.
r|
.
Section 7.2Calculations with Magnitude and Direction