Find the equation of the line which passes through the point $(1,2,3)$ and is parallel to the vector$3\hat i + 2\hat j - 2\hat k$ .

We have, $\vec a = \hat i + 2\hat j + 3\hat k,\vec b = 3\hat i + 2\hat j - 2\hat k$

Vector equation of the line is $\vec a + \lambda \vec b$

$= \hat i + 2\hat j + 3\hat k + \lambda (3\hat i + 2\hat j - 2\hat k)$,

where $\lambda$is any real number.