Exercise 8

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 2.6.22.

Figure 2.6.22 The reflection in the line \(L_\theta\text{.}\)
  1. To do this, notice that this reflection can be obtained by composing three separate transformations as shown in Figure 2.6.23. 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 \(L_\theta\text{.}\)

    Figure 2.6.23 Reflection in the line \(L_\theta\) as a composition of three transformations.

    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