2x2 rotation matrices

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2x2 rotation matrices

A rotation about the origin, anticlockwise, by angle $\theta$ can be expressed in terms of a $2\times 2$ matrix as $$ M_\theta = \left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin\theta & \cos \theta\end{array}\right) $$

This can be seen quite easily by considering the diagram below.

rotation matrix

2x2 rotation matrices 1

Show that the product of two rotation matrices, $M_\theta$ and $M_\phi$ is itself a rotation matrix, $M_{\theta + \phi}$.
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$$\begin{align} M_\phi \times M_\theta &= \left(\begin{array}{cc}\cos \phi & -\sin \phi \\ \sin\phi & \cos \phi\end{array}\right) \left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin\theta & \cos \theta\end{array}\right) \\ &=\left(\begin{array}{cc}\cos \phi\cos\theta-\sin\phi\sin\theta & -\sin \theta\cos\phi -\sin\phi\cos\theta \\ \sin\phi\cos\theta +\cos\phi\sin\theta & \cos \phi\cos\theta -\sin\phi\sin\theta\end{array}\right)\\ &= \left(\begin{array}{cc}\cos (\theta+\phi) & -\sin (\theta + \phi) \\ \sin(\theta +\phi) & \cos (\theta + \phi)\end{array}\right)\\ &= M_{\theta+\phi}\end{align}$$