The line of intersection of the planes $\vec{r}.(3\hat{i} - \hat{j} + \hat{k})=1$ and $\vec{r}. (\hat{i} + 4\hat{j} -2 \hat{k})=2$ is parallel to the vector

$-2\hat{i} +7 \hat{j} + 13\hat{k}$

We know that the line of intersection of the planes $\vec{r}. \vec{n_1}= d_1 $ and $\vec{r}.\vec{n_2}=d_2$ is parallel to $\vec{n_1}×\vec{n_2}$.

Here, $\vec{n_1}= 3\hat{i}- \hat{j}+\hat{k}$ and $\vec{n_2}=\hat{i}+4\hat{j} - 2\hat{k}$

∴ Required vector = $\vec{n_1}×\vec{n_2}=\begin{vmatrix}\hat{i} & \hat{j} & \hat{k}\\3 & -1 & 1\\1 & 4 & -2\end{vmatrix}=-2\hat{i} +7 \hat{j} + 13\hat{k}$