10. In $\triangle A B C$, $A B=\sqrt{2}, A C=\sqrt{3}$, $\angle B A C=30^{\circ}$, and $P$ is any point in the plane of $\triangle A B C$. Then the minimum value of $\mu=\overrightarrow{P A} \cdot \overrightarrow{P B}+\overrightarrow{P B} \cdot \overrightarrow{P C}+\overrightarrow{P C} \cdot \overrightarrow{P A}$ is $\qquad$
10. $\frac{\sqrt{2}}{2}-\frac{5}{3}$.

Notice,
$$
\begin{aligned}
\mu= & \overrightarrow{P A} \cdot \overrightarrow{P B}+\overrightarrow{P B} \cdot \overrightarrow{P C}+\overrightarrow{P C} \cdot \overrightarrow{P A} \\
= & \overrightarrow{P A} \cdot(\overrightarrow{P A}+\overrightarrow{A B}+\overrightarrow{P A}+\overrightarrow{A C})+ \\
& (\overrightarrow{P A}+\overrightarrow{A B}) \cdot(\overrightarrow{P A}+\overrightarrow{A C}) \\
= & 3\left(\overrightarrow{P A}+\frac{\overrightarrow{A B}+\overrightarrow{A C}}{3}\right)^{2}-\frac{1}{3}(\overrightarrow{A B}+\overrightarrow{A C})^{2}+\overrightarrow{A B} \cdot \overrightarrow{A C} .
\end{aligned}
$$
$$
\begin{array}{l}
\text { Then } \mu_{\text {min }}=-\frac{1}{3}(\overrightarrow{A B}+\overrightarrow{A C})^{2}+\overrightarrow{A B} \cdot \overrightarrow{A C} \\
=\frac{\sqrt{2}}{2}-\frac{5}{3} .
\end{array}
$$