Exercise 2.7.39

Start from $Q^{T}Q=I$, as in

(a)

The diagonal entries give $q_1^T{q}_1=1$ and $q_2^T{q}_2=1:$ unit vectors

(b)

The off-diagonal entry is $q_1^T{q}_2=0$ (and in general $q_i^T{q}_j=0)$

(c)

The leading example for $Q$ is the rotation matrix $$\left[\begin{array}{cc}\hfill \cos <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill & \hfill -\sin <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill \\ \hfill \sin <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill & \hfill \cos <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill \end{array}\right].$$