Example 8. For the cube $\mathrm{ABCD}-\mathrm{A}_{1} \mathrm{~B}_{1} \mathrm{C}_{1} \mathrm{D}_{1}$, with edge length $\mathrm{a}$, and the centers of the two bases being $\mathrm{O}, \mathrm{O}_{1}$, find the following distances:
(1) A to $\mathrm{B}_{1} \mathrm{D}_{1}$, (2) $\mathrm{A}$ to $\mathrm{A}_{1} \mathrm{C}$, (3) $\mathrm{A}$ to the plane $\mathrm{BB}_{1} \mathrm{D}_{1} \mathrm{D}$, (4) $\mathrm{A}$ to the plane $\mathrm{A}_{1} \mathrm{BD}$, (5) $\mathrm{BD}$ and $\mathrm{B}_{1} \mathrm{C}$; find the following angles: (6) $\mathrm{BD}_{1}$ and $\mathrm{AC}$, (7) $\mathrm{BD}$ and $\mathrm{CD}_{1}$, (8) $\mathrm{O}_{1} \mathrm{~A}$ and the plane $\mathrm{ABCD}$, (9) the plane $\mathrm{A}_{1} \mathrm{BCD}_{1}$ and the plane $\mathrm{ABCD}$, (10) the plane $\mathrm{D}_{1} \mathrm{AC}$ and the plane $\mathrm{ABCD}$.
(1) $\frac{\sqrt{6}}{2} \mathrm{a}$, (2)
$\frac{\sqrt{6}}{3} \mathrm{a}$,
(3) $\frac{\sqrt{2}}{2} \mathrm{a}$,
(4) $\frac{\sqrt{3}}{3} \mathrm{a}$,
(5) $\frac{\sqrt{3}}{3} \mathrm{a}$,
(6) $90^{\circ}$, (7) $60^{\circ}$, (8) $\operatorname{arctg} \sqrt{2}$, (9)
$45^{\circ},(10) \operatorname{arctg} \sqrt{2}.$