Cartesian Plane Rotated with respect to Another

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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

Rotation-of-cartesian-plane.png


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.

$\blacksquare$


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