Answer
Two different vector parametrizations of the same line:
${{\bf{r}}_1}\left( t \right) = \left( {5,5,2} \right) + t\left( {0, - 2,1} \right)$
${{\bf{r}}_2}\left( t \right) = \left( {5,5,2} \right) + t\left( {0, - 4,2} \right)$
Work Step by Step
Using Eq. (5), the line passes through $P = \left( {5,5,2} \right)$ in the direction of ${\bf{v}} = \left( {0, - 2,1} \right)$ has vector parametrization:
${{\bf{r}}_1}\left( t \right) = \left( {5,5,2} \right) + t\left( {0, - 2,1} \right)$
Let us choose a direction vector ${\bf{w}} = \left( {0, - 4,2} \right)$. Since ${\bf{w}} = 2{\bf{v}}$, the two direction vectors ${\bf{v}}$ and ${\bf{w}}$ are parallel. For ${{\bf{r}}_2}\left( t \right)$ to be another vector parametrization of the same line through $P = \left( {5,5,2} \right)$, we write:
${{\bf{r}}_2}\left( t \right) = \left( {5,5,2} \right) + t\left( {0, - 4,2} \right)$
Thus, ${{\bf{r}}_1}\left( t \right)$ and ${{\bf{r}}_2}\left( t \right)$ are two different vector parametrizations of the same line.
