The equation of the plane containing the lines $\vec{r}=(\hat{i} + \hat{j} - \hat{k})+λ (3\hat{i}-\hat{j})$ and $\vec{r}= (4\hat{i}-\hat{k})+\mu (2\hat{i} + 3 \hat{k}),$ is

$\vec{r}.(3\hat{i} +9 \hat{j} - 2\hat{k})+14 $

Here, $\vec{a_1}= \hat{i} + \hat{j} - \hat{k}, \vec{b_1} = 3\hat{i} - \hat{j},, \vec{a_2} = 4\hat{i} + 0\hat{j} - \hat{k}, \vec{b_1}=3\hat{i}-\hat{j}$ and $\vec{b_2}= 2\hat{i} + 0 \hat{j} + 3\hat{k}.$

The equation of the plane containing the given lines is 

$[\vec{r}\vec{b_1}\vec{b_2}]=[\vec{a_1}\vec{b_1}\vec{b_2}]$

$⇒ \vec{r} .(\vec{b_1}×\vec{b_2})= - 14 $

$⇒ \vec{r} .(-3\hat{i} -9 \hat{j} + 2\hat{k})= -14 $           $[∵ \vec{b_1}×\vec{b_2} = -3\hat{i} -9 \hat{j} + 2\hat{k}]$

$⇒ \vec{r} .(-3\hat{i} +9 \hat{j} - 2\hat{k})=14$