Equation of the plane containing the lines $\overline{\mathrm{r}}=-\overline{\mathrm{i}}+\overline{\mathrm{j}}+2\overline{\mathrm{k}}+\mathrm{\lambda }(-\overline{\mathrm{i}}+\overline{\mathrm{j}}+2\overline{\mathrm{k}}),\overline{\mathrm{r}}=-\overline{\mathrm{i}}+\overline{\mathrm{j}}+2\overline{\mathrm{k}}+\mathrm{\mu }(3\overline{\mathrm{i}}+\overline{\mathrm{k}})\text{ is }$

$$\begin{gathered} \overline{\mathrm{r}}=\overline{\mathrm{a}}+\mathrm{\lambda }\left(\overline{\mathrm{b}}\right), \overline{\mathrm{r}}=\overline{\mathrm{a}}+\mathrm{\mu }\left(\overline{\mathrm{c}}\right) are two lines \\ \overline{\mathrm{b}}\times \overline{\mathrm{c}}=\left|\begin{array}{ccc}\overline{\mathrm{i}}& \overline{\mathrm{j}}& \overline{\mathrm{k}}\\ -1& 1& 2\\ 3& 0& 1\end{array}\right|=\overline{\mathrm{i}}+7\overline{\mathrm{j}}-3\overline{\mathrm{k}} \end{gathered}$$ 

Plane containing the lines is
$$\begin{gathered} (\overline{\mathrm{r}}-\overline{\mathrm{a}})\cdot (\overline{\mathrm{b}}\times \overline{\mathrm{c}})=0 \\ \Rightarrow \overline{\mathrm{r}}\cdot (\overline{\mathrm{b}}\times \overline{\mathrm{c}})=\overline{\mathrm{a}}\cdot (\overline{\mathrm{b}}\times \overline{\mathrm{c}}) \\ \Rightarrow \overline{\mathrm{r}}\cdot (\overline{\mathrm{i}}+7\overline{\mathrm{j}}-3\overline{\mathrm{k}})=0 \end{gathered}$$