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

This is shown in FigureĀ 19.

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Figure2.6.19A sequence of matrix transformations that, when read right to left and top to bottom, rotate a vector about the point \((1,2)\text{.}\)

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

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

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

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

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

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