The equation of a plane containing the lines \(\overline{\mathrm{r}}=(\hat{\imath}+2 \hat{\jmath}-4 \hat{\mathrm{k}})+\lambda(2 \hat{\imath}+3 \hat{\jmath}+6 \hat{\mathrm{k}})\) and
\(\overline{\mathbf{r}}=(\hat{\imath}+3 \hat{\jmath}+4 \hat{\mathrm{k}})+\mu(\hat{\imath}+\hat{\jmath}-\hat{\mathrm{k}})\) is

Normal vector of a plane would be perpendicular to both the given lines and parallel to their cross product.
Now \(\bar{\ell}_{1} \times \bar{\ell}_{2}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & 6 \\ 1 & 1 & -1\end{array}\right|\)
\(=-9 \hat{\mathrm{i}}+8 \hat{\mathrm{j}}-\hat{\mathrm{k}}\) i.e. \(\Rightarrow 9,-8,1\) are d.r. of normal to a plane
Let \(\bar{a}=\hat{i}+3 \hat{j}+4 \hat{k}\) and \(\bar{n}=9 \hat{i}-8 \hat{j}+\hat{k}\)
\(\therefore \overline{\mathrm{r}} \cdot(9 \hat{\mathrm{i}}-8 \hat{\mathrm{j}}+\hat{\mathrm{k}}) =9(1)+(-8)(3)+4 \times 1=9~-\) \(24+4\)
\(\overline{\mathrm{r}} \cdot(9 \hat{\mathrm{i}}-8 \hat{\mathrm{j}}+\hat{\mathrm{k}}) =-11 \Rightarrow 9 \mathrm{x}-8 \mathrm{y}+\mathrm{z}+11=0\) is the equation of plane.