3. In $\triangle A B C$, it is known that $O$ is the circumcenter, the three altitudes $A D, B E, C F$ intersect at point $H$, line $E D$ intersects $A B$ at point $M$, and $F D$ intersects $A C$ at point $N$. Then $\overrightarrow{O H} \cdot \overrightarrow{M N}=$ $\qquad$
3. 0 .

It is known that, $\overrightarrow{O A}+\overrightarrow{O B}+\overrightarrow{O C}=\overrightarrow{O H}$.
Then $\overrightarrow{O H} \cdot \overrightarrow{M N}=(\overrightarrow{O A}+\overrightarrow{O B}+\overrightarrow{O C}) \cdot \overrightarrow{M N}$
$$
\begin{aligned}
= & \overrightarrow{O A} \cdot(\overrightarrow{M F}+\overrightarrow{F E}+\overrightarrow{E N})+ \\
& \overrightarrow{O B} \cdot(\overrightarrow{M F}+\overrightarrow{F N})+\overrightarrow{O C} \cdot(\overrightarrow{M E}+\overrightarrow{E N}) \\
= & (\overrightarrow{O A}+\overrightarrow{O B}) \cdot \overrightarrow{M F}+(\overrightarrow{O A}+\overrightarrow{O C}) \cdot \overrightarrow{E N} .
\end{aligned}
$$

Since $O$ is the circumcenter of $\triangle A B C$, $\overrightarrow{O A}+\overrightarrow{O B}$ and $\overrightarrow{O A}+\overrightarrow{O C}$ pass through the midpoints of segments $A B$ and $A C$, respectively.
Therefore, the value of equation (1) is 0.