Answer
$${\bf{r}}(t)= \langle -2+6t,3t,-2+9t \rangle.$$
Work Step by Step
To find a vector parametrization for the given line, we have to find the direction vector:
$(4,3,7)-(-2,0,-2 )=\langle 6,3,9 \rangle $
Hence, we get the parametrization
$${\bf{r}}(t)= \langle -2,0,-2 \rangle +t \langle 6,3,9 \rangle\\ = \langle -2+6t,3t,-2+9t \rangle.$$
