##### Exercise5

We have seen that the matrix

\begin{equation*} \left[\begin{array}{rr} \cos\theta \amp -\sin\theta \\ \sin\theta \amp \cos\theta \\ \end{array}\right] \end{equation*}performs a rotation through an angle \(\theta\) about the origin. Suppose instead that we would like to rotate by \(90^\circ\) about the point \((1,2)\text{.}\) Using homogeneous coordinates, we will develop a matrix that performs this operation.

Our strategy is to

begin with a vector whose tail is at the point \((1,2)\text{,}\)

translate the vector so that its tail is at the origin,

rotate by \(90^\circ\text{,}\) and

translate the vector so that its tail is back at \((1,2)\text{.}\)

This is shown in FigureĀ 19.

Remember that, when working with homogeneous coordinates, we consider matrices of the form

\begin{equation*} \left[\begin{array}{rrr} a \amp b \amp c \\ d \amp e \amp f \\ 0 \amp 0 \amp 1 \\ \end{array}\right] \text{.} \end{equation*}The first operation is a translation by \((-1,-2)\text{.}\) Find the matrix that performs this translation.

The second operation is a \(90^\circ\) rotation about the origin. Find the matrix that performs this rotation.

The third operation is a translation by \((1,2)\text{.}\) Find the matrix that performs this translation.

Use these matrices to find the matrix that performs a \(90^\circ\) rotation about \((1,2)\text{.}\)

Use your matrix to determine where the point \((-10, 5)\) ends up if rotated by \(90^\circ\) about the \((1,2)\text{.}\)