3. If the complex number $z$ satisfies $|z+\mathrm{i}|+|z-2-\mathrm{i}|=2 \sqrt{2}$, then the minimum value of $|z|$ is $\qquad$ .
3. $\frac{\sqrt{2}}{2}$.

The given expression can be rewritten as
$$
\begin{array}{l}
|(z-1)-(-1-\mathrm{i})|+|(z-1)-(1+\mathrm{i})| \\
=2 \sqrt{2} .
\end{array}
$$

By geometric interpretation, the minimum value of $|z|$ is the distance from the point $(0,0)$ to the line segment $y=x-1(0 \leqslant x \leqslant 2)$, which is $\frac{\sqrt{2}}{2}$.