Answer
The meteor hits the ground at $t=4$ minutes.
Work Step by Step
The parametrization of the trajectory is ${\bf{r}}\left( t \right) = \left( {2,1,4} \right) + t\left( {3,2, - 1} \right)$ km.
In component forms, we have
$x=2+3t$, ${\ \ }$ $y=1+2t$, ${\ \ }$ $z=4-t$
Since the surface of the earth is represented by the $xy$-plane, the meteor hits the ground when $z=0$. So,
$z=4-t=0$, ${\ \ }$ $t=4$.
So, the meteor hits the ground at $t=4$ minutes.
