1.1. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ with edge $a$. Find the angle and distance between the lines $A_{1} B$ and $A C_{1}$.
1.1. To prove that the triangle $A_{1} B D$ is equilateral. Moreover, point $A$ is equidistant from its vertices. Therefore, it projects onto the center of this triangle. Similarly, point $C_{1}$ projects onto the center of triangle $A_{1} B D$. Consequently, the lines $A_{1} B$ and $A C_{1}$ are perpendicular, and the distance between them is equal to the distance from the center of triangle $A_{1} B D$ to its side. Since the sides of this triangle are equal to $a \sqrt{2}$, the required distance is $a / \sqrt{\overline{6}}$.