Answer
A vector parametrization for the line is
${\bf{r}}\left( t \right) = \left( {1 + 2t,1 + t,1 + 4t} \right)$,
where $ - \infty < t < \infty $.
Work Step by Step
A line through the points $\left( {2,0, - 1} \right)$ and $\left( {4,1,3} \right)$ has direction vector ${\bf{v}}$ given by
${\bf{v}} = \left( {4,1,3} \right) - \left( {2,0, - 1} \right) = \left( {2,1,4} \right)$
We write ${P_0} = \left( {1,1,1} \right)$. So, ${{\bf{r}}_0} = \overrightarrow {O{P_0}} = \left( {1,1,1} \right)$.
Using Eq. (5), the line through ${P_0} = \left( {1,1,1} \right)$ in the direction of ${\bf{v}} = \left( {2,1,4} \right)$ has vector parametrization:
${\bf{r}}\left( t \right) = {{\bf{r}}_0} + t{\bf{v}} = \left( {1,1,1} \right) + t\left( {2,1,4} \right)$
${\bf{r}}\left( t \right) = \left( {1 + 2t,1 + t,1 + 4t} \right)$,
where $ - \infty < t < \infty $.
