Exercise 6.B.1

Proof. (a) One can easily check that each of the four vectors has norm ${\sin <!--\; FUNCTION\; APPLICATION\; -->}^2𝜃+{\cos <!--\; FUNCTION\; APPLICATION\; -->}^2𝜃$, which equals $1$. Moreover, we have

which shows that they are orthogonal.

(b) Clearly, for any $v$ and $u$ in ${ℝ}^2$ with $\left|\right|v\left|\right|=\left|\right|u\left|\right|=1$, we can write $v=(\cos <!--\; FUNCTION\; APPLICATION\; -->𝜃,\sin <!--\; FUNCTION\; APPLICATION\; -->𝜃)$ and $v=(\cos <!--\; FUNCTION\; APPLICATION\; -->\alpha ,\sin <!--\; FUNCTION\; APPLICATION\; -->\alpha )$ for some angles $𝜃$ and $\alpha$. If $v,u$ is an orthonormal basis, then we must have

One solution is to take choose $𝜃$ and $\alpha$ such that $𝜃-\alpha =\frac{\pi }{2}$. Then

This shows that $v,u$ is of the first form given in part (a). □