Example 1 As shown in Figure 1, two equilateral triangles $\triangle ABC$ and $\triangle A_{1} B_{1} C_{1}$ (vertices arranged clockwise) are placed on a plane, and the midpoints $D$ of sides $BC$ and $B_{1} C_{1}$ coincide. Try to find:
(1) the angle between $A A_{1}$ and $B B_{1}$; (2) the ratio $A A_{1}: B B_{1}$.
\begin{array}{l}\text { Sol: } \because B B_{1} \xrightarrow{S\left(D, 90^{\circ}, \sqrt{3}\right)} A A_{1}, \\ \therefore A A_{1} \perp B B_{1} \text { and } A A_{1}: B B_{1}=\sqrt{3} .\end{array}