Answer
The parametric equations are: $x=1-t$, $y=-1$, $z=2t$, where $ - \infty < t < \infty $.
Work Step by Step
A line passing through the points $\left( {1, - 1,0} \right)$ and $\left( {0, - 1,2} \right)$ has direction vector ${\bf{v}}$ given by
${\bf{v}} = \left( {0, - 1,2} \right) - \left( {1, - 1,0} \right) = \left( { - 1,0,2} \right)$
We choose the point ${P_0} = \left( {1, - 1,0} \right)$. So, ${{\bf{r}}_0} = \overrightarrow {O{P_0}} = \left( {1, - 1,0} \right)$.
Using Eq. (5), the line passes through ${P_0} = \left( {1, - 1,0} \right)$ in the direction of ${\bf{v}} = \left( { - 1,0,2} \right)$ has vector parametrization:
${\bf{r}}\left( t \right) = {{\bf{r}}_0} + t{\bf{v}} = \left( {1, - 1,0} \right) + t\left( { - 1,0,2} \right)$
${\bf{r}}\left( t \right) = \left( {1 - t, - 1,2t} \right)$
So, the parametric equations are: $x=1-t$, $y=-1$, $z=2t$, where $ - \infty < t < \infty $.
