Vector equation of a line passing through a point whose position vector is $\overrightarrow a$ and parallel to a given vector is

$\overrightarrow {r}=\overrightarrow {a}+ \lambda \overrightarrow {b}$

The given point is

$\overrightarrow a=\hat i-2 \hat j+3 \hat k$

The given parallel vector is

$\overrightarrow b=3\hat i+2 \hat j+6 \hat k$

Therefore equation of the line is

$\overrightarrow r=(\hat i-2 \hat j+3 \hat k)+\lambda (3 \hat i+2 \hat j+ 6 \hat k)$