Exercise 10

This exercise is about $2\times2$ orthogonal matrices.

1. In The geometry of matrix transformations, we saw that the matrix $\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$ represents a rotation by an angle $\theta$. Explain why this matrix is an orthogonal matrix.

2. We also saw that the matrix $\begin{bmatrix} \cos\theta & \sin\theta \\ \sin\theta & -\cos\theta \end{bmatrix}$ represents a reflection in a line. Explain why this matrix is an orthogonal matrix.

3. Suppose that $\uvec_1=\twovec{\cos\theta}{\sin\theta}$ is a 2-dimensional unit vector. Use a sketch to indicate all the possible vectors $\uvec_2$ such that $\uvec_1$ and $\uvec_2$ form an orthonormal basis of $\real^2$.

4. Explain why every $2\times2$ orthogonal matrix is either a rotation or a reflection.