##### 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 Figure 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*}
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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