Rotations and General Reflections - Questions
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Find the matrix for each of the following rotations about the origin
i) \(60^\circ\) anticlockwise
ii) \(225^\circ\) clockwise
i) \(\left[ {\begin{array}{*{20}{c}}{\dfrac{1}{2}}&{ - \dfrac{{\sqrt 3 }}{2}}\\{\dfrac{{\sqrt 3 }}{2}}&{\dfrac{1}{2}}\end{array}} \right]\)
ii) \(\left[ {\begin{array}{*{20}{c}}{ - \dfrac{1}{{\sqrt 2 }}}&{ - \dfrac{1}{{\sqrt 2 }}}\\{\dfrac{1}{{\sqrt 2 }}}&{ - \dfrac{1}{{\sqrt 2 }}}\end{array}} \right]\)
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$$\begin{align}&\text{(i)}\quad
M=\left[\begin{array}{cc}
\cos 60^{\circ}-\sin 60^{\circ} \\
\sin 60^{\circ} & \cos 60^{\circ}
\end{array}\right]=\left[\begin{array}{cc}
\frac{1}{2} & -\frac{\sqrt{3}}{2} \\
\frac{\sqrt{3}}{2} & \frac{1}{2}
\end{array}\right]\\
&\begin{aligned}\text{(ii)}\quad M &=\left[\begin{array}{c}\cos \left(-225^{\circ}\right) & -\sin \left(-225^{4}\right) \\ \sin \left(-225^{\circ}\right) & \cos \left(-225^{\circ}\right)\end{array}\right] \\ &=\left[\begin{array}{c}\cos \left(225^\circ+\sin\left(225^{\circ}\right)\right. \\ -\sin \left(225^{\circ}\right)+\cos \left(125^{\circ}\right)\end{array}\right] \\ &=\left[\begin{array}{l}-\cos 45^{\circ}-\sin 45^{\circ} \\ +\sin 45^{\circ}-\cos 45^{\circ}\end{array}\right]\\
&=\left[\begin{array}{c}-\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}\end{array}\right] \end{aligned}
\end{align}$$
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