3. For a square $A B C D$ with side length 4, if it is folded along $B D$ to form a $60^{\circ}$ dihedral angle, then the distance between the midpoint of $B C$ and $A$ is $\qquad$ .
3. $2 \sqrt{2}$.

Take the midpoint $O$ of $B D$, it is easy to prove that $\triangle A C O$ is an equilateral triangle with side length $2 \sqrt{2}$, so $A C=2 \sqrt{2}$. Let the midpoint of $B C$ be $M$, then in $\triangle A C B$, we have
$$
A M=\sqrt{\frac{1}{2}\left(A C^{2}+A B^{2}\right)-\frac{1}{4} B C^{2}}=2 \sqrt{2} .
$$