Equations of plane passing through (1, 0, 2) and line of intersection of planes

$\overline{\mathrm{r}}.\left(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}\right) = 1$ and $\overline{\mathrm{r}}.\left(\hat{\mathrm{i}}- 2\widehat{ \mathrm{j}}\right) = -2$ is 

$$\begin{gathered} \overline{\mathrm{r}}.(\mathrm{i}+\mathrm{j}+\mathrm{k})-1+\mathrm{\lambda }(\overline{\mathrm{r}}.(-\mathrm{i}+2\mathrm{j})-2)=0 \\ \mathrm{passes} \mathrm{through} (1, 0, 2) \\ 1+2-1+\mathrm{\lambda }(-1-2)=0 \Rightarrow \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right. \\ \overline{\mathrm{r}}.(3\mathrm{i}+3\mathrm{j}+3\mathrm{k})-3+\overline{\mathrm{r}}.(-2\mathrm{i}+4\mathrm{j})-4=0 \\ \overline{\mathrm{r}}.(\mathrm{i}+3\mathrm{j}+3\mathrm{k})=7 \end{gathered}$$