A vector parallel to the line of intersection of the planes \(\bar{r} \cdot(3 \hat{i}-\hat{j}+\hat{k})=1\) and \(\overline{\mathrm{r}} \cdot(\hat{\mathrm{i}}+4 \hat{\mathrm{j}}-2 \hat{\mathrm{k}})=2\) is

The line of intersection of the planes \(\overline{\mathrm{r}} \cdot(3 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})=1\) and \(\overline{\mathrm{r}} \cdot(\hat{\mathrm{i}}+4 \hat{\mathrm{j}}-2 \hat{\mathrm{k}})=2\) is perpendicular to each of the normal vectors \(\overline{n_1}=3 \hat{i}-\hat{j}+\hat{k}\) and \(\overline{n_2}=\hat{i}+4 \hat{j}-2 \hat{k}\).
\(\therefore \quad\) The line is parallel to the vector \(\overline{\mathrm{n}}_1 \times \overline{\mathrm{n}}_2\)
\(\begin{aligned}
\therefore \quad \overline{\mathrm{n}}_1 \times \overline{\mathrm{n}}_2 & =\left|\begin{array}{ccc}
\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\
3 & -1 & 1 \\
1 & 4 & -2
\end{array}\right| \\
& =-2 \hat{\mathrm{i}}+7 \hat{\mathrm{j}}+13 \hat{\mathrm{k}}
\end{aligned}\)