Answer
The two beams intersect if $c=10$.
Work Step by Step
The two laser's beam intersect if there exist parameter values $t$ and $s$ such that
$\left( {1,2,4} \right) + t\left( {2,1, - 1} \right) = \left( {6,3, - 1} \right) + s\left( { - 5,2,c} \right)$
In component forms, we have
$x=1+2t=6-5s$, ${\ \ }$ $y=2+t=3+2s$, ${\ \ }$ $z = 4 - t = - 1 + sc$
Solving the first two equations we obtain $s = \frac{1}{3}$, $t = \frac{5}{3}$. Substituting these values in the third equation reconciles the $z$-coordinates:
$z = 4 - \frac{5}{3} = - 1 + \frac{1}{3}c$
So we obtain $c=10$. Thus, the two beams intersect if $c=10$.
