# Rotation in a Cartesian Plane

## Rotation in a Cartesian Plane

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

### Formulas

• The rule of a rotation $$r_O$$ of 90° centered on the origin point $$O$$ of the Cartesian plane, in the positive direction (counter-clockwise), is $$r_O : (x, y) ↦ (−y, x)$$.
The rule of a rotation $$r_O$$ of 180° centered on the origin point $$O$$ of the Cartesian plane, in the positive direction (counter-clockwise) is $$r_O : (x, y) ↦ (−x, −y)$$.
The rule of a rotation $$r_O$$ of 270° centered on the origin point $$O$$ of the Cartesian plane in the positive direction (counter-clockwise), is $$r_O : (x, y) ↦ (y, −x)$$.
• For a rotation $$r_O$$ of 90° centered on the origin point $$O$$ of the Cartesian plane, the transformation matrix is $$\begin{bmatrix}0 & −1\\1 & 0\end{bmatrix}$$, so that the coordinates $$(x’, y’)$$ of a point $$P(x, y)$$ by this rotation will be given by $$\begin{bmatrix}0 & −1\\1 & 0\end{bmatrix}\times \begin{bmatrix}y \\x\end{bmatrix}=\begin{bmatrix}x’\\y’\end{bmatrix}$$.
• For a rotation $$r_O$$ of 180° centered on the origin point $$O$$ of the Cartesian plane, the transformation matrix is $$\begin{bmatrix}−1 & 0\\0 & −1\end{bmatrix}$$, so that the coordinates $$(x’, y’)$$ of a point $$P(x, y)$$ by this rotation will be 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 rotation $$r$$ of $$theta°$$ centered on the origin point in a Cartesian plane, the transformation matrix is $$\begin{bmatrix}\cos{\theta} & −\sin{\theta}\\\sin{\theta} & \cos{\theta}\end{bmatrix}$$, so that the coordinates $$(x’, y’)$$ of a point $$P(x, y)$$ by this rotation will be given by $$\begin{bmatrix}\cos{\theta} & −\sin{\theta}\\\sin{\theta} & \cos{\theta}\end{bmatrix}\times \begin{bmatrix}x \\y\end{bmatrix}=\begin{bmatrix}x’\\y’\end{bmatrix}$$.

### Example

Here is the Cartesian representation of a rotation of 90° centered on the origin point:

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

For example, for a rotation of 90° of the point (−3,6) around the origin point: $$\begin{bmatrix}-1×6\\1×-3\end{bmatrix}=\begin{bmatrix}-6\\-3\end{bmatrix}$$