Example. The edge length of the cube $A C^{\prime}$ is $a$, find the distance between $B D$ and $A B^{\prime}$.
Solution One: As shown in Figure 1, where $O E \perp A O^{\prime}$, thus $O E \perp$ plane $A B^{\prime} D^{\prime}$. But $B D / A$ plane $A B^{\prime} D^{\prime}$, so the length of $O E$ is the required distance. (Calculation omitted)

Solution Two: As shown in Figure 2, let $E$ be a point on $B D$, with $B E = x$. Draw $E G \perp D B$ intersecting $A B$ at $G$, and draw $F G \perp A B$, intersecting $A B^{\prime}$ at $F$. Then $G B = \sqrt{2} x, G F = A G = a - \sqrt{2} x$. Therefore,
$$
E F^{2} = x^{2} + (a - \sqrt{2})^{2}
$$
$= 3\left(a - \frac{\sqrt{2}}{8} a\right)^{2} + \frac{a^{2}}{3}$, when $x = \frac{\sqrt{2}}{8} a$, $E F$ reaches its minimum value $\frac{\sqrt{3}}{3} a$, which is the distance between the skew lines.
(Author's unit: Wuhan Yiyuan No.1 High School)