TITLE: Use complex numbers to prove that the composition of rotations is another rotation
QUESTION [7 upvotes]: From ACOPS by Paul Zeitz, q4.2.30, a section on complex numbers.
"Let $R_a(\theta)$ denote the transformation of the plane that rotates everything about the center point $a$ by $\theta$ radians counterclockwise. Prove the interesting fact that the composition of $R_a(\theta)$ and $R_b(\phi)$ is another rotation $R_c(\alpha)$. Find $c,\alpha$ in terms of $a,b,\theta,\phi$."
I've tried to argue that $R_a(\theta)$ maps the point $z$ on the complex plane to $z^\prime = e^{i\theta}(z-a)+a,$ and therefore the composition of $R_a(\theta)$ and $R_b(\phi)$ maps $z$ to $z^\prime = e^{i\phi}(e^{i\theta}(z-a)+a-b)+b.$ I have then tried to write this expression in the form $e^{i(\phi+\theta)}(z-c)+c$ but have been unsuccessful.

REPLY [5 votes]: You need to solve
$$e^{i\phi}(e^{i\theta}(z-a)+a-b)+b=e^{i(\phi+\theta)}(z-c)+c$$ for $c$.
Rearranging the terms, you get
$$e^{i(\phi+\theta)}z+e^{i\phi}\left(\left(1-e^{i\theta}\right)a-b\right))+b=e^{i(\phi+\theta)}z+c\left(1-e^{i(\phi+\theta)}\right),$$
which validates the claim about composition and yields
$$c=\frac{e^{i\phi}\left(\left(1-e^{i\theta}\right)a-b\right))+b}{1-e^{i(\phi+\theta)}}.$$