3. $\sqrt{2}$.

Take the midpoint $E$ of $A B$, then $D E / / A C, \angle A D E=90^{\circ}, \angle D A E=\angle A-90^{\circ}$. Let $A C=b, A D=m$, then $\tan C=\frac{m}{b}, \tan \left(A-90^{\circ}\right)=\frac{b}{2 m}$, i.e., $-\cot A=\frac{b}{2 m}$. Therefore,
$$
\tan C-\cot A=\frac{m}{b}+\frac{b}{2 m} \geqslant \sqrt{2},
$$

with equality if and only if $m=\frac{\sqrt{2}}{2} b$.
3. In $\triangle A B C$, $D$ is the midpoint of side $B C$. If $\overrightarrow{A D} \cdot \overrightarrow{A C}=0$, then the minimum value of $\tan C-\cot A$ is
$\qquad$.