Chapter 4.3, Problem 61E

Rotations If a point $\left(x,y\right)$ in the plane is rotated counterclockwise about the origin through an angle of 45°, its new coordinates $\left(x^{\prime },y^{\prime }\right)$ are given by

$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=R\left[\begin{array}{c}x\\ y\end{array}\right]$ where $R$ is the $2\times 2$ matrix $\left[\begin{array}{cc}a& -a\\ a& a\end{array}\right]\text{ and }a=\sqrt{1/2}\approx 0.7071$.

a. If the point $\left(2,3\right)$ is rotated counterclockwise through an angle of 45°, what are its (approximate) new coordinates?

b. Multiplication by what matrix would result in a counterclockwise rotation of 90°? 135°? (Express the matrices in terms of R.) [HinT: Think of a rotation through 90° as two successive rotations through 45°.]

c. Multiplication by what matrix would result in a clockwise rotation of 45°?