# 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.

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