Example 11 If the angle (defined as the angle between the positive directions of the two lines) between two skew lines $a$ and $b$ is $\theta$ ($0<\theta<\pi$), take points $A', E, A, F$ on lines $a$ and $b$ respectively. If the positive directions of $A'E$ and $AF$ are the same as those of $a$ and $b$, and $\angle EA'A=\alpha, \angle A'AF=\beta, A'A=m$, then the distance between $a$ and $b$ is
$$
d=\frac{m}{\sin \theta} \sqrt{1-\cos ^{2} \theta-\cos ^{2} \alpha-\cos ^{2} \beta-2 \cos \alpha \cdot \cos \beta \cdot \cos \theta} .
$$
Prove: As shown in Fig. $24-9$, draw $A E^{\prime} / / A^{\prime} E$, then in the trihedral angle $A-$ $F E^{\prime} A^{\prime}$, $\angle E^{\prime} A F=\theta, \angle A^{\prime} A E^{\prime}=\pi-\alpha$. By property $5, A^{\prime} A$ and the plane $E^{\prime} A F$ form an angle $\tau_{A^{\prime} A}$ such that $\sin \tau_{R^{\prime} A}=\sqrt{S(A)} / \sin \theta$, and the distance from $A^{\prime}$ to the plane $E^{\prime} A F$ is the distance $d$ between $a$ and $b$, hence
$$
\begin{aligned}
d & =m \cdot \sin \tau_{A^{\prime} A} . \\
& =\frac{m}{\sin \theta} \cdot \sqrt{1-\cos ^{2} \theta-\cos ^{2} \alpha-\cos ^{2} \beta-2 \cos \alpha \cdot \cos \beta \cdot \cos \theta} .
\end{aligned}
$$