# Cartesian Plane Rotated with respect to Another

## Theorem

Let $\mathbf r$ be a position vector embedded in a Cartesian plane $\CC$ with origin $O$.

Let $\CC$ be rotated anticlockwise through an angle $\varphi$ about the axis of rotation $O$.

Let $\CC'$ denote the Cartesian plane in its new position.

Let $\mathbf r$ be kept fixed during this rotation.

Let $\tuple {x, y}$ denote the components of $\mathbf r$ with respect to $\CC$.

Let $\tuple {x', y'}$ denote the components of $\mathbf r$ with respect to $\CC'$.

Then:

 $\ds x'$ $=$ $\ds x \cos \varphi + y \sin \varphi$ $\ds y'$ $=$ $\ds -x \sin \varphi + y \cos \varphi$

## Proof

Let $\mathbf r$ be represented by a directed line segment whose initial point coincides with the origin $O$.

Let the terminal point of $\mathbf r$ be identified with the point $P$.

Let $\CC$ be rotated to $\CC'$ through an angle $\varphi$ as shown, keeping $P$ fixed.

We have that:

 $\ds x'$ $=$ $\ds OA + BP$ by inspection $\ds$ $=$ $\ds x \cos \varphi + y \sin \varphi$ Definition of Cosine, Definition of Sine $\ds y'$ $=$ $\ds xB - xA$ by inspection $\ds$ $=$ $\ds y \cos \varphi - x \sin \varphi$ Definition of Cosine, Definition of Sine

hence the result.

$\blacksquare$