# Reflection in a Cartesian Plane

## Reflection in a Cartesian Plane

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

### Formulas

• The rule for a reflection $$r_x$$ over the x-axis in a Cartesian plane is $$s_x : (x, y) ↦ (x, −y)$$. The rule for a reflection $$r_y$$ over the y-axis in a Cartesian plane is $$r_y : (x, y) ↦ (−x, y)$$.
• For a reflection $$r_x$$ over the x-axis in a Cartesian plane, the transformation matrix is $$\begin{bmatrix}1 & 0\\0 & −1\end{bmatrix}$$, such that the coordinates $$(x’, y’)$$ of a point $$P(x, y)$$ under this reflection are given by $$\begin{bmatrix}1 & 0\\0 & −1\end{bmatrix}\times \begin{bmatrix}x \\y\end{bmatrix}=\begin{bmatrix}x’\\y’\end{bmatrix}$$.
• For a reflection $$r_x$$ over the y-axis in a Cartesian plane, the transformation matrix is $$\begin{bmatrix}−1 & 0\\0 & 1\end{bmatrix}$$, such that the coordinates $$(x’, y’)$$ of a point $$P(x, y)$$ under the reflection are given by $$\begin{bmatrix}-1 & 0\\0 & 1\end{bmatrix}\times \begin{bmatrix}x \\y\end{bmatrix}=\begin{bmatrix}x’\\y’\end{bmatrix}$$.

### Example

This is a Cartesian representation of a reflection over the x-axis:

The definition of this reflection may be written as: $$r_x : (x, y) ↦ (x, −y)$$ or, in matrix form, as: $$\begin{bmatrix}1 & 0\\0 &−1\end{bmatrix}\times \begin{bmatrix}x \\y\end{bmatrix}=\begin{bmatrix}x’\\y’\end{bmatrix}$$.

For example, for the reflection of the point $$(-5,3)$$ : $$\begin{bmatrix}1×−5\\−1 ×3\end{bmatrix}=\begin{bmatrix}−5\\−3\end{bmatrix}$$