general rotations

Question

Let $R$ be the rotation about the point $(1,0)$ by an angle of $45$ degrees.

By using matrix methods:

Find the image of the line $2x-3y+1=0$ under $R$

I would really appreciate it if someone could help me with this. Thanks.

Answer 1

You can work in the complex plane, where making rotations around the origin is easy. Let $a+ib$ the center of the rotation and $\theta$ its angle.

Take a point $z=x+iy\in \mathbb C$. We need to make a translation to rotate $z$ around the origin (we will multiply by $e^{i\theta}$) and then make the inverse translation to put things in their place. Then, $$\begin{align} x+iy&\longrightarrow(x-a)+(y-b)i\\ &\longrightarrow[(x-a)\cos\theta-(y-b)\sin\theta]+i[(x-a)\sin\theta+(y-b)\cos\theta]\\ &\longrightarrow[(x-a)\cos\theta-(y-b)\sin\theta+a]+i[(x-a)\sin\theta+(y-b)\cos\theta+b] \end{align}$$

With matrix notation, we have: $$\left(\begin{array}{c}x\\y\end{array}\right)\rightarrow \left(\begin{array}{cc}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{array}\right) \left[ \left(\begin{array}{c}x\\y\end{array}\right)-\left(\begin{array}{c}a\\b\end{array}\right)\right]+ \left(\begin{array}{c}a\\b\end{array}\right) $$

To apply the rotation to the line, you'd better write its equation in parametric form.