Answer
There is no solution, so the lines do not intersect.
Work Step by Step
Suppose the lines intersect and there is point of intersection at $t$ and $s$ such that ${{\bf{r}}_1}\left( t \right) = {{\bf{r}}_2}\left( s \right)$. So,
$\left( {2,1,1} \right) + t\left( { - 4,0,1} \right) = \left( { - 4,1,5} \right) + s\left( {2,1, - 2} \right)$
In component forms, we have
$x = 2 - 4t = - 4 + 2s$, ${\ \ }$ $y = 1 = 1 + s$, ${\ \ }$ $z = 1 + t = 5 - 2s$
Solving the first two equations, we get $s=0$ and $t = \frac{3}{2}$. However, these values do not satisfy the third equation since they do not have the same $z$-coordinates. It is a contradiction. Therefore, there has no solution and the lines do not intersect.
