A, B, C are points in the plane. (CA, CB) = (pi)/(3) and CA = 2CB. Let r be the rotation with center C and angle (pi)/(3). Find the image of A under r.

Here's a step-by-step solution to your geometry problem:

Let $C$ be the origin $(0,0)$ in the complex plane. Let $c=0$.
The rotation $r$ is centered at $C$ with angle $\frac{\pi}{3}$. If $z'$ is the image of $z$ under $r$, then $z' = e^{i\pi/3} z$.

Given:

1. $(\vec{CA}, \vec{CB}) = \frac{\pi}{3} \pmod{2\pi}$.
2. $CA = 2CB$.
3. $r$ is the rotation with center $C$ and angle $\frac{\pi}{3}$.

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