1.4. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ with edge $1, K$ - the midpoint of edge $D D_{1}$. Find the angle and the distance between the lines $C K$ and $A_{1} D$.
1.4. Let's calculate the angle between the lines $C$ and $A_{1} D$. Let $M$ be the midpoint of edge $B B_{1}$. Then $A_{1} M \| K C$, so the angle between the lines $C$ and $A_{1} D$ is equal to the angle $M A_{1} D$. This angle can be calculated using the cosine theorem, since $A_{1} D=\sqrt{2}, A_{1} M=\sqrt{5 / 2}$ and $D M=32$. After some calculations, we obtain $\cos M A_{1} D=1 / \sqrt{10}$.

To calculate the distance between the lines $C K$ and $A_{1} D$, we project them onto the plane passing through the edges $A B$ and $C_{1} D_{1}$. The line $A_{1} D$ is projected to the midpoint $O$ of segment $A D_{1}$, and points $C$ and $K$ are projected to the midpoints $Q$ of segment $B C_{1}$ and the midpoint $P$ of segment $O D_{3}$. The distance between the lines $C K$ and $A_{1} D$ is equal to the distance from point $O$ to the line $P Q$. The legs $O P$ and $O Q$ of the right triangle $O P Q$ are equal to $1 / \sqrt{8}$ and 1. Therefore, its hypotenuse is $3 / \sqrt{8}$. The desired distance is equal to the product of the lengths of the legs divided by the length of the hypotenuse, i.e., it is equal to $1 / 3$.