Example 1. In the cube $A B C D$ $-\mathrm{A}_{1} \mathrm{~B}_{1} \mathrm{C}_{\mathrm{i}} \mathrm{D}_{1}$, find the distance between $A B_{1}$ and $\mathrm{A}_{1} \mathrm{C}_{1}$.
Solve As shown in Figure 2, the dihedral angle $A-C_{1} A_{1}-B_{1}$ has a plane angle of $90^{\circ}, \mathrm{A}$ is at a distance of $\mathrm{a}$ from $\mathrm{A}_{1} \mathrm{C}_{1}, \mathrm{B}_{1}$ is at a distance of $\frac{\sqrt{2}}{2} \mathrm{a}$ from $\mathrm{A}_{1} \mathrm{C}_{1}$, and by formula (2), the distance $\mathrm{h}=\frac{\sqrt{3}}{3} \mathrm{a}$ between the skew lines $\mathrm{AB}_{1}$ and $\mathrm{A}_{1} \mathrm{C}_{1}$.