Rectangular Cartesian coordinates – iGCSE Further Pure Mathematics Edexcel Revision

Understanding Rectangular Cartesian Coordinates

The Rectangular Cartesian Coordinate System is a two or three-dimensional system in which the position of a point is specified by its distances along perpendicular lines from reference points.
A point is defined by a pair of numerical coordinates (x, y) in two dimensions and a triple of numerical coordinates (x, y, z) in three dimensions.
The reference points are known as the origins of the coordinates which is denoted as (0,0) in 2D and (0,0,0) in 3D.
The perpendicular lines are referred to as axes, where the horizontal line is the x-axis, the vertical line is the y-axis, and in three dimensions, the third axis (usually coming out from the paper or screen) is the z-axis.

Plotting a point in the Cartesian system involves identifying its coordinates and marking that point on the plane or in space.
The coordinates (x, y) or (x, y, z) are read along the axes from the origin.
For example, the point (2, 3) in 2D is found by moving 2 units along the x-axis and 3 units up the y-axis from the origin.

In the 2D Cartesian plane, the x and y axes divide the plane into four regions called quadrants.
Quadrants are numbered counterclockwise from the upper right. The upper right quadrant is the first quadrant, the upper left is the second, the lower left is the third, and the lower right is the fourth.
In the first quadrant, both x and y coordinates are positive. In the second quadrant, x is negative and y is positive. In the third quadrant, x and y are both negative. In the fourth quadrant, x is positive and y is negative.

The distance between two points in the Cartesian system can be calculated using Pythagoras’ theorem for 2D, and a derivative of that theorem for 3D.
For two points (x1, y1) and (x2, y2), the distance is (√[(x2-x1)² + (y2-y1)²]).
For points in 3D, an additional z coordinate is included ([√[(x2-x1)² + (y2-y1)² + (z2-z1)²]).

Unlike the Cartesian coordinates, which use perpendicular lines to locate a point, the Polar coordinates system uses the distance from a fixed point and an angle from a fixed direction.
Polar coordinates are especially useful when dealing with problems involving circular symmetry while Cartesian coordinates are more convenient for describing standard space and plane problems.