On the coordinate plane, $O$ is the origin, and points $A(1,0)$ and $B(-2,0)$ are given. There are also two points $P$ and $Q$ in the upper half-plane satisfying $\overline{AP} = \overline{OA}$, $\overline{BQ} = \overline{OB}$, $\angle POQ$ is a right angle. Let $\angle AOP = \theta$.
If $\sin\theta = \frac{3}{5}$, find the coordinates of point $Q$ and explain that $\overrightarrow{BQ} = 2\overrightarrow{AP}$. (Non-multiple choice question, 6 points)