10. In parallelogram $O A D B$, $\overrightarrow{O A}=\vec{a}, \overrightarrow{O B}=\vec{b}$. Also, $M$ and $N$ are on line segments $A B$ and $C D$ respectively, and $B M=\frac{1}{3} B C, C N=\frac{1}{3} C D$, where $C$ is the intersection of diagonals $A B$ and $O D$. Express $\overrightarrow{M N}=$ $\qquad$ in terms of $\vec{a}$ and $\vec{b}$.
$$
\begin{array}{l}
\text { 10. } \frac{1}{2} \vec{a}-\frac{1}{6} \vec{b} . \\
\begin{aligned}
\overrightarrow{M C} & =\frac{2}{3} \overrightarrow{B C}=\frac{1}{3} \overrightarrow{B A}=\frac{1}{3}(\vec{a}-\vec{b}), \overrightarrow{C N}=\frac{1}{3} \overrightarrow{C D}=\frac{1}{6} \overrightarrow{O D} \\
& =\frac{1}{6}(\vec{a}+\vec{b}), \overrightarrow{M N}=\overrightarrow{M C}+\overrightarrow{C N} .
\end{aligned}
\end{array}
$$
10. $\frac{1}{2} \vec{a}-\frac{1}{6} \vec{b}$.