Vector Equation of a line passing through the point with position vector $\overrightarrow a$
and parallel to the given vector $\overrightarrow b$ is
$\overrightarrow r = \overrightarrow a + t.\overrightarrow b$
$i.e.,\:\: \overrightarrow r =( \hat i + 2\hat j + 3\hat k) + t(3\hat i+2\hat j- 2\hat k)$
The catesian equation is $\large\frac{x-x_1}{l}=\frac{y-y_1}{m}=\frac{ z-z_1}{n}$
That is $\large\frac{x-1}{3}= \frac{y-2}{2} = \frac{z+2}{-2}$
