Answer
$${\bf{r}} = \left( { - {\bf{i}} + 2{\bf{j}}} \right) + t\left( {2{\bf{i}} + \frac{3}{4}{\bf{j}}} \right)$$
Work Step by Step
$$\eqalign{
& {\bf{r}}\left( t \right) = \left( {2t - 1} \right){\bf{i}} + \sqrt {3t + 4} {\bf{j}};{\text{ }}{P_0}\left( { - 1,2} \right) \cr
& {\text{Let }}t = 0,{\text{ then }}{\bf{r}}\left( 0 \right) \cr
& {\bf{r}}\left( 0 \right) = \left( {2\left( 0 \right) - 1} \right){\bf{i}} + \sqrt {3\left( 0 \right) + 4} {\bf{j}} \cr
& {\bf{r}}\left( 0 \right) = - {\bf{i}} + 2{\bf{j}} \cr
& t = 0{\text{ at }}{P_0} \cr
& {\text{Calculate }}{\bf{r}}{\text{'}}\left( 0 \right) \cr
& {\bf{r}}'\left( t \right) = \frac{d}{{dt}}\left[ {\left( {2t - 1} \right){\bf{i}} + \sqrt {3t + 4} {\bf{j}}} \right] \cr
& {\bf{r}}'\left( t \right) = 2{\bf{i}} + \frac{3}{{2\sqrt {3t + 4} }}{\bf{j}} \cr
& {\bf{r}}'\left( 0 \right) = 2{\bf{i}} + \frac{3}{{2\sqrt {3\left( 0 \right) + 4} }}{\bf{j}} \cr
& {\bf{r}}'\left( 0 \right) = 2{\bf{i}} + \frac{3}{4}{\bf{j}} \cr
& {\text{The vector equation is}} \cr
& {\bf{r}} = \left( { - {\bf{i}} + 2{\bf{j}}} \right) + t\left( {2{\bf{i}} + \frac{3}{4}{\bf{j}}} \right) \cr} $$
