University Mathematics Handbook (2015)
III. Analytic Geometry and Vectors
Chapter 1. Coordinate Systems in the Plane
1.1 Cartesian Coordinate System in a Plane
A Cartesian coordinate system consists of two number lines perpendicular to each other, where the horizontal is 
and the vertical is 
, intersecting at a point called the origin, and dividing the plane into four quarters. See Figure 1.
Each point 
in the plane is specified by an ordered pair of numbers 
, where “
” stands for the distance of from the -axis, and marked with 
(plus) if is right of the -axis, or with 
(minus) if s left of the -axis. “
” is the distance of from 
-axis, and is marked with (plus) if is above the -axis, or with (minus) if is below the -axis. Thus, pair is the coordinates of point .
Figure 1 shows the coordinates of 
, 
, 
, and 
.
1.2 Polar Coordinate System
Let's fix point 
in the plane, called a pole, and a ray 
. The position of point in the plane is strictly defined by its distance 
from pole and the positive angle 
between vector 
and axis 
, when taken from to, counterclockwise.
The pair 
are called the polar coordinates of point .
All points in the plane have a unique pair of polar coordinates. Where and 
, Pole is represented by 
and any angle , which means the pole is not uniquely represented.
Figure 1 Figure 2 Figure 3
The polar coordinates of the points in Figure 2 are:
1.3 Relation Between Cartesian and Polar Coordinates
Fixing Pole at the origin of Cartesian system, and axis on its -axis, we get the relations between the two coordinate systems (Figure 3).
And, vice versa, if and are known, the result should be
1.4 Distance Between Two Points 
1.5 Area of a Triangle with Vertices
