Coordinate Systems in the Plane - Analytic Geometry and Vectors - University Mathematics Handbook

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