Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector.

It is given that the line passes through the point $A(1,2,3)$. Therefore, the position vector through $A$ is $\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}$

$\vec{b}=3 \hat{i}+2 \hat{j}-2 \hat{k}$\

It is known that the line which passes through point $\mathrm{A}$ and parallel to $\vec{b}$ is given by $\vec{r}=\vec{a}+\lambda \vec{b}$, where $\lambda$ is a constant.

$\Rightarrow \vec{r}=\hat{i}+2 \hat{j}+3 \hat{k}+\lambda(3 \hat{i}+2 \hat{j}-2 \hat{k})$

This is the required equation of the line.