Vector Rotation and Reflection
Let $ \mathbf{v}_0 $ be a vector. The vector $ \mathbf{v}_0 $ is rotated about the origin by an angle of $ 42^\circ $ counter-clockwise, taking it to vector $ \mathbf{v}_1 $. The vector $ \mathbf{v}_1 $ is then reflected over the line with direction vector $ \begin{pmatrix} \cos 108^\circ \\ \sin 108^\circ \end{pmatrix}, $ taking it to vector $ \mathbf{v}_2 $.
The vector $ \mathbf{v}_2 $ can also be produced by reflecting the vector $ \mathbf{v}_0 $ over the line with direction vector $ \begin{pmatrix} \cos \theta \\ \sin \theta \end{pmatrix}, $ where $ \theta $ is an acute angle. Find $ \theta $.
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Solution
The matrix for the rotation is given by
\[\begin{pmatrix} \cos 42^\circ & -\sin 42^\circ \\ \sin 42^\circ & \cos 42^\circ \end{pmatrix}.\]In general, the matrix for reflecting over the line with direction vector $ \begin{pmatrix} \cos \theta \\ \sin \theta \end{pmatrix} $ is given by $ \begin{pmatrix} \cos 2 \theta & \sin 2 \theta \\ \sin 2 \theta & -\cos 2 \theta \end{pmatrix} $. So here, the matrix for the reflection is
\[\begin{pmatrix} \cos 216^\circ & \sin 216^\circ \\ \sin 216^\circ & -\cos 216^\circ \end{pmatrix}.\]
Hence, the matrix that takes $ \mathbf{v}_0 $ to $ \mathbf{v}_2 $ is
\begin{align*}
&\quad \ \begin{pmatrix} \cos 216^\circ & \sin 216^\circ \\ \sin 216^\circ & -\cos 216^\circ \end{pmatrix} \begin{pmatrix} \cos 42^\circ & -\sin 42^\circ \\ \sin 42^\circ & \cos 42^\circ \end{pmatrix} \\
&= \begin{pmatrix} \cos 216^\circ \cos 42^\circ + \sin 216^\circ \sin 42^\circ & -\cos 216^\circ \sin 42^\circ + \sin 216^\circ \cos 42^\circ \\ \sin 216^\circ \cos 42^\circ - \cos 216^\circ \sin 42^\circ & -\sin 216^\circ \sin 42^\circ - \cos 216^\circ \cos 42^\circ \end{pmatrix} \\
&= \begin{pmatrix} \cos (216^\circ - 42^\circ) & \sin (216^\circ - 42^\circ) \\ \sin (216^\circ - 42^\circ) & -\cos (216^\circ - 42^\circ) \end{pmatrix} \\
&= \begin{pmatrix} \cos 174^\circ & \sin 174^\circ \\ \sin 174^\circ & -\cos 174^\circ \end{pmatrix}.\end{align*}Thus, $ \theta = 174^\circ/2 = \boxed{87^\circ} $.
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