Find the matrix associated with the rotations (a) \( \pi / 2 \) (b) \( \pi / 4 \) (c) \( \pi / 6 \).

To find the rotation matrices associated with the angles \( \pi/2 \), \( \pi/4 \), and \( \pi/6 \), we can use the standard rotation matrix formula for a counterclockwise rotation by an angle \( \theta \): \[ R(\theta) = \begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix} \] Now, we will compute the rotation matrices for each specified angle. ### (a) Rotation by \( \pi/2 \) For \( \theta = \pi/2 \): \[ R\left(\frac{\pi}{2}\right) = \begin{pmatrix} \cos\left(\frac{\pi}{2}\right) & -\sin\left(\frac{\pi}{2}\right) \\ \sin\left(\frac{\pi}{2}\right) & \cos\left(\frac{\pi}{2}\right) \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \] ### (b) Rotation by \( \pi/4 \) For \( \theta = \pi/4 \): \[ R\left(\frac{\pi}{4}\right) = \begin{pmatrix} \cos\left(\frac{\pi}{4}\right) & -\sin\left(\frac{\pi}{4}\right) \\ \sin\left(\frac{\pi}{4}\right) & \cos\left(\frac{\pi}{4}\right) \end{pmatrix} = \begin{pmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{pmatrix} \] ### (c) Rotation by \( \pi/6 \) For \( \theta = \pi/6 \): \[ R\left(\frac{\pi}{6}\right) = \begin{pmatrix} \cos\left(\frac{\pi}{6}\right) & -\sin\left(\frac{\pi}{6}\right) \\ \sin\left(\frac{\pi}{6}\right) & \cos\left(\frac{\pi}{6}\right) \end{pmatrix} = \begin{pmatrix} \frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{pmatrix} \] ### Summary of Rotation Matrices - For \( \pi/2 \): \[ R\left(\frac{\pi}{2}\right) = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \] - For \( \pi/4 \): \[ R\left(\frac{\pi}{4}\right) = \begin{pmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{pmatrix} \] - For \( \pi/6 \): \[ R\left(\frac{\pi}{6}\right) = \begin{pmatrix} \frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{pmatrix} \]