Polar Coordinates
A vector in the plane can be identified by its x- and
(os y-coordinates.
(e’), (e)) For
example the vector (2, 3) is the vector whose x-coordinate is 2 and whose
y-coordinate is 3. These are called Cartesian coordinates.
There is another type of coordinates that are commonly used to identify
vectors in the plane. They’re called polar coordinates. We’ll discuss what
polar coordinates are in this chapter. Roughly speaking, instead of identifying
vectors with their x- and y-coordinates, we’ll identify them with their distance
from the origin and a point on the unit circle which tells us which direction
the vector is from the origin.
Scalar multiplication of vectors
Recall that if r 2 R and if (a, b) 2 R2 (that is, if r is a scalar and (a, b) is
a vector), then
r(a, b) = (ra, rb)
For example, 2(3, 5) = (6, 10).
If r > 0, then the vector r(a, b) points in the same direction from the origin
as the vector (a, b). If (a, b) is a vector whose norm equals 1, then r(a, b) is
a vector whose norm is 1 scaled by r. That is, the norm of r(a, b) is r times
1, which equals r. Conversely, if (c, d) is a vector whose norm equals r, then
1
1
r (c, d) is a vector whose norm equals r times r, which is 1.
r (G b)
Vectors on the unit circle
The points on the unit circle are the vectors whose norm equals 1. The
points on the unit circle are also those vectors that can be written in the form
( cos(✓), sin(✓) ) for some number ✓.
243
(e))
%—
0
0
(os (e’),
What polar coordinates are
-ç
-÷
0
0
n
‘
Any point in the plane can be identified by its distance from the origin—
which is the norm of the vector—and the direction one would have to travel
from the origin to reach the point. Notice that the direction from the origin
can be given as a point on the unit circle.
r (G b)
These two pieces of information taken together—a distance r 0 from the
origin, and a point ( cos(✓), sin(✓) ) on the unit circle—are the two “coordinates” used to determine the position of a point in the plane when using
polar coordinates. That is, any point in the plane can be given as a point of
the form
r( cos(✓), sin(✓) )
where r 0 and ( cos(✓), sin(✓) ) is a point on the circle.
A disadvantage of polar coordinates
Polar coordinates can be very useful, as we’ll see. They do have a disadvantage to Cartesian coordinates though in that it can be difficult to add two
vectors if they are written in polar coordinates.
244
How to write a Cartesian vector in polar coordinates
%—
0
0
If we are given a vector (a, b) in Cartesian coordinates, then we can also
write it in polar coordinates. Here’s how:
Let r 0 be the norm of the vector (a, b). We can scale the vector (a, b)
by the scalar 1r , and we’d have a new vector 1r (a, b) that points in the same
direction as our original vector, but its distance from the origin is 1. That is,
the vector ( ar , rb ) = 1r (a, b) is a vector on the unit circle, so it’s a point of the
form ( cos(✓), sin(✓) ), as any vector on the unit circle is.
What we have is that
-ç
-÷
0
0
n
‘
⇣a b ⌘
(a, b) = r ,
r r
Notice that the vector is written in Cartesian coordinates on the left side of
the equal sign, and on the right side it is written in polar coordinates. Because
( ar , rb ) is a vector on the unit circle, it’s a vector of the form ( cos(✓), sin(✓) ),
even if it doesn’t quite look like it.
To summarize,
If r
0 is the norm of the vector (a, b),
⇣a b ⌘
then
,
is a point on the unit circle
r r
⇣a b ⌘
and r ,
is the vector (a, b) written in polar coordinates.
r r
245
Problem. Write the vector ( 3, 7) in polar coordinates. That is, write
( 3, 7) as r( cos(✓), sin(✓) ) for some r 0 and some point on the unit circle
( cos(✓), sin(✓) ).
Solution.
We’llpuse the formula above. The norm of the vector ( 3, 7) is
p
2
( 3) + 72 = 58. Thus, we can write ( 3, 7) in polar coordinates as
p ⇣ 3
7 ⌘
58 p , p
58
58
3
The point ( p58
, p758 ) is on the unit circle, so it’s an example of a vector of
the form ( cos(✓), sin(✓) ), as any vector on the unit circle is.
/
U
246
Exercises
In #1-6, match the vectors written in polar coordinates with the vectors
drawn to the right. Remember that the norm, or length, of the vector
r cos(✓), sin(✓) is r. The vector cos(✓), sin(✓) is the point in the unit
circle that represents the direction of the vector r cos(✓), sin(✓) .
2⇡
cos( 2⇡
3 ), sin( 3 )
3.)
5⇡
cos( 5⇡
4 ), sin( 4 )
P1
2.)
a.,
1.) 3 cos( ⇡4 ), sin( ⇡4 )
4.) 2 cos( ⇡4 ), sin( ⇡4 )
⇡
6 ), sin(
⇡
6)
0>
5.) 3 cos(
2⇡
6.) 2 cos( 2⇡
3 ), sin( 3 )
Write the vectors in #7-11 in polar coordinates. That is, write each vector
in the form r( cos(✓), sin(✓) ) for some r 0 and some point in the unit circle
( cos(✓), sin(✓) ).
7.) (2, 1)
8.) ( 12 , 18)
10.) (9, 4)
11.) (
9.) ( 13 , 15 )
p
3 1
2 , 2)
For #12-18, identify the given integer.
12.) loge (e3 )
13.) loge (e2 )
14.) loge (e)
16.) loge ( 1e )
17.) loge ( e12 )
18.) loge ( e13 )
247
15.) loge (1)