Chapter 5.5, Problem 4E

Let $\theta$ be a fixed real number and let
$x_1=\left(\begin{array}{c}\cos \theta \\ \sin \theta \end{array}\right)$ and $x_2=\left(\begin{array}{c}-\sin \theta \\ \cos \theta \end{array}\right)$

Show that $\left\{x_1,x_2\right\}$ is an orthonormal basis for ${ℝ}^2.$

Given a vector y in ${ℝ}^2,$ write it as a linear combination $c_1{x}_1+c_2{x}_2.$

Verify that $c_1^2+c_2^2={\Vert y\Vert }^2=y_1^2+y_2^2$