7. Let $G$ and $O$ be the centroid and circumcenter of $\triangle A B C$, respectively, and $A B=\sqrt{3}, A C=3$. Then $\overrightarrow{A G} \cdot \overrightarrow{A O}=$
$$
\overrightarrow{A G} \cdot \overrightarrow{A O}=\frac{1}{3}(\overrightarrow{A B}+\overrightarrow{A C}) \cdot \overrightarrow{A O}=\frac{1}{6}\left(\overrightarrow{A B}^{2}+\overrightarrow{A C}^{2}\right)=2
$$