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.
$\blacksquare$
Sources
- 1970: George Arfken: Mathematical Methods for Physicists (2nd ed.) ... (next): Chapter $1$ Vector Analysis $1.2$ Rotation of Coordinates
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): rotation