3. Given a rectangular cuboid $A B C D-A_{1} B_{1} C_{1} D_{1}$, where $A D=1, A B=2, A A_{1}=3$, point $P$ is a point within the plane $A_{1} B D$. Then the minimum length of $A P$ is
As shown in the figure, draw $A E \perp B D$ at point $E$, connect $A_{1} E$, draw $A P \perp$ $A_{1} E$ at point $P$, then $B D \perp A A_{1} \Rightarrow B D \perp$ plane $A_{1} A E$ $\Rightarrow B D \perp A P \Rightarrow A P \perp$ plane $A_{1} B D$.
Then $A E=\frac{2}{\sqrt{5}} \Rightarrow A_{1} E=\sqrt{\frac{4}{5}+9}=\frac{7}{\sqrt{5}}$
$\Rightarrow \sin \angle A_{1} E A=\frac{3 \sqrt{5}}{7} \Rightarrow A P_{\min }=A E \sin \angle A_{1} E A$
$=\frac{2}{\sqrt{5}} \cdot \frac{3 \sqrt{5}}{7}=\frac{6}{7}$.