11. In the complex plane, non-zero complex numbers $z_{1}, z_{2}$ lie on a circle centered at $i$ with a radius of 1, the real part of $\bar{z}_{1} \cdot z_{2}$ is zero, and the principal value of the argument of $z_{1}$ is $\frac{\pi}{6}$, find $z_{2}$.
11. Since $\arg z_{1}=\frac{\pi}{6}, z_{1}$ makes an angle of $\frac{\pi}{3}$ with the $y$-axis, then $\left|z_{1}\right|=\left|O_{1} Z_{1}\right|=1$, and $z_{1}=$ $\frac{\sqrt{3}}{2}+\frac{1}{2} i$. Also, the real part of $\bar{z}_{1} \cdot z_{2}$ is 0, so $\arg z_{2}-\frac{\pi}{6}=\frac{\pi}{2}$ or $\frac{3 \pi}{2}$. Since $Z_{2}$ lies on the circle $|z-i|=1$, we have $\arg z_{2}=\frac{2 \pi}{3}$, which means $\angle Z_{2} O Z_{1}=\frac{\pi}{2}$, and $Z_{2}$ lies on the diameter passing through $Z_{1}$ and the center of the circle, $\left|z_{2}\right|$ $=\sqrt{3}$. Therefore, $z_{2}=-\frac{\sqrt{3}}{2}+\frac{3}{2} i$.