1. Let $O$ and $H$ be the circumcenter and orthocenter of $\triangle ABC$, respectively. Please derive the complex number corresponding to the orthocenter $H$ with $O$ as the origin of the complex plane.
1. Let the complex numbers corresponding to the three vertices of a triangle be $z_{1}, z_{2}, z_{3}$, and the complex number corresponding to $H$ be $z$. Then $\frac{z-z_{1}}{z_{3}-z_{2}}, \frac{z-z_{2}}{z_{3}-z_{1}}$ are both purely imaginary, so $\frac{z-z_{1}}{z_{3}-z_{2}}=-\frac{\bar{z}-\bar{z}_{1}}{\bar{z}_{3}-\bar{z}_{2}}, \frac{z-z_{2}}{z_{3}-z_{1}}=-\frac{\bar{z}-\bar{z}_{2}}{\bar{z}_{3}-\bar{z}_{1}}$.
Eliminating $\bar{z}$ and combining with $z_{1} \bar{z}_{1}=z_{2} \bar{z}_{2}=z_{3} \bar{z}_{3}=1$, we get $z=z_{1}+z_{2}+z_{3}$, which is the complex number corresponding to the orthocenter.