9. For the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with an edge length of 1, the distance between the lines $A_{1} C_{1}$ and $B D_{1}$ is $\qquad$
9. $\frac{\sqrt{6}}{6}$

As shown in Figure 6, construct the section $B B_{1} D_{1} D$ of the cube, then $\Lambda_{1} C_{1}$. In the plane $B B_{1} D_{1} D$, let $\Lambda_{1} C_{1}$ intersect $B_{1} D_{1}$ at point $O$. Within the plane $B B_{1} D_{1} D$, construct $O H \perp B D_{1}$, with $H$ being the foot of the perpendicular. Then $O H$ is the common perpendicular of $A_{1} C_{1}$ and $B D_{1}$. Clearly, $O H$ is equal to half the altitude of the hypotenuse of $\mathrm{R}_{1} \triangle B B_{1} \mathrm{D}_{1}$, i.e., $O H=\frac{\sqrt{6}}{6}$.