Simple Nature - Light and Matter

l/Example 59.

one point and ends at another. Adding two ∆rvectors is interpreted
as a trip with two legs: by computing the ∆rvector going from point
A to point B plus the vector from B to C, we find the vector that
would have taken us directly from A to C.

Calculations with magnitude and direction
If you ask someone where Las Vegas is compared to Los Angeles,
she is unlikely to say that the ∆xis 290 km and the ∆yis 230 km,
in a coordinate system where the positivexaxis is east and they
axis points north. She will probably say instead that it’s 370 km
to the northeast. If she was being precise, she might specify the
direction as 38◦counterclockwise from east. In two dimensions, we
can always specify a vector’s direction like this, using a single angle.
A magnitude plus an angle suffice to specify everything about the
vector. The following two examples show how we use trigonometry
and the Pythagorean theorem to go back and forth between thex-y
and magnitude-angle descriptions of vectors.

Finding magnitude and angle from components example 59

.Given that the∆rvector from LA to Las Vegas has∆x=290 km

and∆y=230 km, how would we find the magnitude and direction

of∆r?

.We find the magnitude of∆rfrom the Pythagorean theorem:

|∆r|=

√

∆x^2 +∆y^2

= 370 km

We know all three sides of the triangle, so the angleθcan be

found using any of the inverse trig functions. For example, we

know the opposite and adjacent sides, so

θ= tan−^1

∆y

∆x

= 38◦.

Finding the components from the magnitude and angle example

60

.Given that the straight-line distance from Los Angeles to Las

Vegas is 370 km, and that the angleθin the figure is 38◦, how

can thexandycomponents of the∆rvector be found?

.The sine and cosine of θrelate the given information to the

information we wish to find:

cosθ=

∆x

|∆r|

sinθ=

∆y

|∆r|

Section 3.4 Motion in three dimensions 201