# Cartesian Coordinate System

**A two-dimensional coordinate system in which every point is uniquely identified by a pair of signed distances to a pair of perpendicular lines.**

![[CartesianCoordinateSystem.svg]]
_A Cartesian coordinate plane_

The **Cartesian coordinate system** is a two-dimensional coordinate system that uniquely identifies each point in two-dimensional Euclidean space using an _ordered pair of signed distances_ to an ordered pair of fixed perpendicular lines known as the _axes_.

The intersection of the two axes is known as the _origin_ and is defined as .

## Higher dimensions

The Cartesian coordinate system can be generalised to _any_ -dimensional Euclidean space.

In an -dimensional Euclidean space, points are specified with an ordered set of _signed distances_ to an ordered set of fixed perpendicular _dimension hyperplanes_ instead.

![[CartesianCoordinateSystem3D.svg]]
_The Cartesian coordinate system in three dimensions with the - Cartesian coordinate plane embedded_

## Notation

In two-dimensional Cartesian coordinates, the horizontal and vertical axis are often denoted as the _-axis_ and the _-axis_, respectively. Extending into three dimensions, the third axis is often denoted the _-axis_.

The coordinates themselves are referred to, in order, as the , , and coordinates. The and coordinates may also be referred to as the _abscissa_ and the _ordinate_, respectively. Generic coordinates are thus notated as or .

The axes and coordinates may also be denoted using _subscripts_, as in , which is extendable to any -dimensions.