Translation in a Cartesian Plane

Transformation of

$\mathbb{R} \times \mathbb{R}$ in

$\mathbb{R} \times \mathbb{R}$ whose Cartesian representation corresponds to a translation of the geometric plane.

Formulas

The rule of a translation $t$ with a vector $(a, b)$ in a Cartesian plane is $t_{a, b} : (x, y) ↦ (x + a, y + b)$ .
For a translation $t$ in the Cartesian plane that is defined by a vector $\overrightarrow{t}(a, b)$ , the transformation matrix is $\begin{bmatrix}x + a\\y+b\end{bmatrix}$ , such that the coordinates $(x’, y’)$ of a point $P(x, y)$ after the translation will be given by $\begin{bmatrix}x + a\\y + b\end{bmatrix}=\begin{bmatrix}x’\\y’\end{bmatrix}$ .

Example

This is the Cartesian representation of a translation t by a vector (5, 1).

The definition of this translation may be written as: $t_{5, 1} : (x, y) ↦ (x + 5, y + 1)$ or, in matrix form: $\begin{bmatrix}x + 5\\y + 1\end{bmatrix}=\begin{bmatrix}x’\\y’\end{bmatrix}$

For example, for the translation of the point $(-3,1)$ : $\begin{bmatrix}-3 + 5\\1 + 1\end{bmatrix}=\begin{bmatrix}2\\2\end{bmatrix}$