Answer
The two lines intersect.
Intersection point: $\left( {\frac{5}{4},\frac{{11}}{2}} \right)$.
Work Step by Step
The two lines intersect if there exist parameter values $t$ and $s$ such that
$\left( { - 1,1} \right) + t\left( {2,4} \right) = \left( {2,1} \right) + s\left( { - 1,6} \right)$
In component forms, we have
$ - 1 + 2t = 2 - s$, ${\ \ \ }$ $1 + 4t = 1 + 6s$
Solving the two equations we obtain $t = \frac{9}{8}$ and $s = \frac{3}{4}$.
Substituting $t = \frac{9}{8}$ in ${{\bf{r}}_1}\left( t \right) = \left( { - 1,1} \right) + t\left( {2,4} \right)$ gives the intersection point: ${{\bf{r}}_1}\left( {\frac{9}{8}} \right) = \left( {\frac{5}{4},\frac{{11}}{2}} \right)$.
Likewise, substituting $s = \frac{3}{4}$ in ${{\bf{r}}_2}\left( s \right) = \left( {2,1} \right) + s\left( { - 1,6} \right)$ also gives the same intersection point: ${{\bf{r}}_2}\left( {\frac{3}{4}} \right) = \left( {\frac{5}{4},\frac{{11}}{2}} \right)$.
