##### Exercise8

We saw earlier that the rotation in the plane through an angle $$\theta$$ is given by the matrix:

\begin{equation*} \left[\begin{array}{rr} \cos\theta \amp -\sin\theta \\ \sin\theta \amp \cos\theta \\ \end{array}\right] \text{.} \end{equation*}

We would like to find a similar expression for the matrix that represents the reflection in $$L_\theta\text{,}$$ the line passing through the origin and making an angle of $$\theta$$ with the positive $$x$$-axis, as shown in Figure 22.

1. To do this, notice that this reflection can be obtained by composing three separate transformations as shown in Figure 23. Beginning with the vector $$\xvec\text{,}$$ we apply the transformation $$R$$ to rotate by $$-\theta$$ and obtain $$R(\xvec)\text{.}$$ Next, we apply $$S\text{,}$$ a reflection in the horizontal axis, followed by $$T\text{,}$$ a rotation by $$\theta\text{.}$$ We see that $$T(S(R(\xvec)))$$ is the same as the reflection of $$\xvec$$ in the original line $$L_\theta\text{.}$$

Using this decomposition, show that the reflection in the line $$L_\theta$$ is described by the matrix

\begin{equation*} \left[\begin{array}{rr} \cos(2\theta) \amp \sin(2\theta) \\ \sin(2\theta) \amp -\cos(2\theta) \\ \end{array}\right] \text{.} \end{equation*}

You will need to remember the trigonometric identities:

\begin{equation*} \begin{aligned} \cos(2\theta) \amp {}={} \cos^2\theta - \sin^2\theta \\ \sin(2\theta) \amp {}={} 2\sin\theta\cos\theta \\ \end{aligned} \text{.} \end{equation*}
2. Now that we have a matrix that describes the reflection in the line $$L_\theta\text{,}$$ show that the composition of the reflection in the horizontal axis followed by the reflection in $$L_\theta$$ is a counterclockwise rotation by an angle $$2\theta\text{.}$$ We saw some examples of this earlier in Exercise 2.

in-context