Cartesian Coordinate System

Cartesian Coordinates

Cartesian coordinates are the simplest and most commonly used coordinates. They identify each point (vector) in the Euclidean space ${\mathbb{R}}^n$ by its components. Essentially, the $k$-th Cartesian coordinate of $\mathbf{p}$ is its $k$-th component.

Theorem: Cartesian Coordinate System

The function $x:{\mathbb{R}}^n\to {\mathbb{R}}^n$ defined for each $\mathbf{p}=\left[\begin{array}{ccc}{p}_1& \cdots & p_n\end{array}\right]\in {\mathbb{R}}^n$ as

$$x(\mathbf{p})\stackrel{\text{def}}{=}(x^1,\dots ,x^n),$$

where $x^k=p_k$ ($k=1,\dots ,n$) is a coordinate system for ${\mathbb{R}}^n$.

PROOF

TODO

NOTATION

We denote coordinates using superscripts instead of subscripts, i.e. $x^1,\dots ,x^n$ instead of $x_1,\dots ,x_n$, in order to make it clear that we are talking about coordinates and not components. However, since the values of a point’s Cartesian coordinates coincide with the values of its components, you might sometimes see the Cartesian coordinates labelled using subscripts.

Notation: Cartesian Coordinates in 2D and 3D

Cartesian coordinates in ${\mathbb{R}}^2$ are usually denoted as $x,y$ and in ${\mathbb{R}}^3$ as $x,y,z$.