Translation in a Cartesian Plane

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.


  • 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 \(\begin{bmatrix}x + a\\y + b\end{bmatrix}=\begin{bmatrix}x’\\y’\end{bmatrix}\).


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 \((-3,1)\) : \(\begin{bmatrix}-3 + 5\\1 + 1\end{bmatrix}=\begin{bmatrix}2\\2\end{bmatrix}\)

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