11.3. All edges of a regular triangular prism $A B C A_{1} B_{1} C_{1}$ have length $a$. Points $M$ and $N$ lie on the lines $B C_{1}$ and $C A_{1}$, respectively, and the line $M N$ is parallel to the plane $A A_{1} B$. What is the minimum length of such a segment $M N$?
11.3. If $M^{\prime}$ and $N^{\prime}$ are the projections of points $M$ and $N$ onto the plane $A B C$, then $M^{\prime} N^{\prime} \| A B$. Let $C M^{\prime}=x$. Then $M^{\prime} N^{\prime}=x$, and the length of the projection of segment $M N$ onto line $C C_{1}$ is $|a-2 x|$. Therefore,

$$
M N^{2}=x^{2}+(a-2 x)^{2}=5 x^{2}-4 a x+a^{2}
$$

The minimum length of segment $M N$ is $a / \sqrt{5}$.