Matrix Equation of Plane Rotation
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Theorem
Let $r_\alpha$ be a rotation of the plane about the origin through an angle of $\alpha$.
Let $r_\alpha$ rotate an arbitrary point in the plane $P = \tuple {x, y}$ onto $P' = \tuple {x', y'}$
Then:
- $\begin {bmatrix} x' \\ y' \end {bmatrix} = \begin {bmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end {bmatrix} \begin {bmatrix} x \\ y \end {bmatrix}$
Proof
Let the coordinates of $P'$ be encoded as the elements of a $2 \times 1$ matrix.
We have:
| \(\ds \begin {bmatrix} x' \\ y' \end {bmatrix}\) | \(=\) | \(\ds P'\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \map {r_\alpha} P\) | Definition of Plane Rotation | |||||||||||
| \(\ds \) | \(=\) | \(\ds \begin {bmatrix} x \cos \alpha - y \sin \alpha \\ x \sin \alpha + y \cos \alpha \end {bmatrix}\) | Equation defining Plane Rotation | |||||||||||
| \(\ds \) | \(=\) | \(\ds \begin {bmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end {bmatrix} \begin {bmatrix} x \\ y \end {bmatrix}\) | Definition of Matrix Product (Conventional) |
Hence the result.
$\blacksquare$