Find the vector equation of the line passing through the point $2i+3j+k$ and parallel to the vector $4i−2j+3k$.

The correct answer is $(2 + 4\lambda) \hat{i}+ (3 - 2 \lambda) \hat{j}+ (1+ 3 \lambda) \hat{k}$

Given that the line passes through the point $\overrightarrow{a}$ = $2i+3j+k$

It is parallel to line $\overrightarrow{b}$ = $4i−2j+3k$

The vector equation of the line passing through a point and parallel to the given line as

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

= $2i+3j+k$ + $\lambda (4i−2j+3k)$

= $(2 + 4\lambda)i+ (3 - 2 \lambda)j+ (1+ 3 \lambda)k$