find vector equation of plane passing through point  whose position vector is 5i+2j-3k and perpendicular to intersection of planes r.(2i-j+2k)=0 and r.(i+3j-5k)+7=0

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$$\begin{gathered} \mathrm{Let} \mathrm{plane} \mathrm{be} \overrightarrow{\mathrm{r}}.\overrightarrow{\mathrm{n}}=\overrightarrow{\mathrm{a}}.\overrightarrow{\mathrm{n}}, \mathrm{where} \overrightarrow{\mathrm{n}} \mathrm{is} \mathrm{normal} \mathrm{vector}. \\ \mathrm{As} \mathrm{line} \mathrm{of} \mathrm{intersection} \mathrm{is} \mathrm{perpendicular} \mathrm{of} \mathrm{plane}. \mathrm{Hence} \mathrm{direction} \mathrm{ratio} \\ \mathrm{of} \mathrm{line} \mathrm{will} \mathrm{be} \mathrm{normal} \mathrm{of} \mathrm{plane}. \\ \overrightarrow{\mathrm{r}}.\left(2\hat{\mathrm{i}}-\hat{\mathrm{j}}+2\hat{\mathrm{k}}\right)=0 \\ \overrightarrow{{\mathrm{n}}_1}=2\hat{\mathrm{i}}-\hat{\mathrm{j}}+2\hat{\mathrm{k}} \\ \overrightarrow{\mathrm{r}}.\left(\hat{\mathrm{i}}+3\hat{\mathrm{j}}-5\hat{\mathrm{k}}\right)+7=0 \\ \overrightarrow{{\mathrm{n}}_2}=\hat{\mathrm{i}}+3\hat{\mathrm{j}}-5\hat{\mathrm{k}} \\ \mathrm{Hence} \mathrm{normal} \mathrm{of} \mathrm{plane} \mathrm{is} \\ \overrightarrow{\mathrm{n}}=\overrightarrow{{\mathrm{n}}_1}\times \overrightarrow{{\mathrm{n}}_2} \\ =\left|\begin{array}{ccc}\hat{\mathrm{i}}& \hat{\mathrm{j}}& \hat{\mathrm{k}}\\ 2& -1& 2\\ 1& 3& -5\end{array}\right| \\ =\left(5-6\right)\hat{\mathrm{i}}-\left(-10-2\right)\hat{\mathrm{j}}+\left(6+1\right)\hat{\mathrm{k}} \\ \overrightarrow{\mathrm{n}}=-\hat{\mathrm{i}}+12\hat{\mathrm{j}}+7\hat{\mathrm{k}} \\ \mathrm{Given} \mathrm{that} \\ \overrightarrow{\mathrm{a}}=5\hat{\mathrm{i}}+2\hat{\mathrm{j}}-3\hat{\mathrm{k}} \\ \mathrm{Hence} \mathrm{equation} \mathrm{is} \\ \overrightarrow{\mathrm{r}}.\overrightarrow{\mathrm{n}}=\overrightarrow{\mathrm{a}}.\overrightarrow{\mathrm{n}} \\ \overrightarrow{\mathrm{r}}.\left(-\hat{\mathrm{i}}+12\hat{\mathrm{j}}+7\hat{\mathrm{k}}\right)=\left(-\hat{\mathrm{i}}+12\hat{\mathrm{j}}+7\hat{\mathrm{k}}\right).\left(5\hat{\mathrm{i}}+2\hat{\mathrm{j}}-3\hat{\mathrm{k}}\right) \\ =-5+24-21 \\ =-2 \\ \Rightarrow \overrightarrow{\mathrm{r}}.\left(-\hat{\mathrm{i}}+12\hat{\mathrm{j}}+7\hat{\mathrm{k}}\right)=-2 \\ \mathbf{\Rightarrow }\overrightarrow{\mathbf{r}}\mathbf{.}\left(\mathbf{-}\hat{\mathbf{i}}\mathbf{+}\mathbf{12}\hat{\mathbf{j}}\mathbf{+}\mathbf{7}\hat{\mathbf{k}}\right)\mathbf{+}\mathbf{2}\mathbf{=}\mathbf{0} \end{gathered}$$

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