Official solution
P1 : 2x + 3y + 2z = 0
\[\Rightarrow\] \[{\overrightarrow n _1} = 2\widehat i + 3\widehat j + 2\widehat k\]
P2 : x - 2y + z = 0
\[\Rightarrow\] \[{\overrightarrow n _2} = \widehat i - 2\widehat j + \widehat k\]
Direction vector of line L which is line of intersection of P1 \& P2
\[\overrightarrow r = {\overrightarrow n _1} \times {\overrightarrow n _2} = 7\widehat i - 7\widehat k\]
DR's of L are (1, 0, -1)
\[\Rightarrow\] Equation of L : \[{x \over 1} = {y \over 0} = {z \over { - 1}} = \lambda \]
DR's of \[\overrightarrow {PQ} \] = (\[\lambda\] + 1, -2, 2 - \[\lambda\])
\[\because\] \[\overrightarrow {PQ} \bot \overrightarrow r \]
\[ \Rightarrow (\lambda + 1)(1) + ( - 2)(0) + (2 - \lambda )( - 1) = 0\]
\[ \Rightarrow \lambda = {1 \over 2} \Rightarrow Q\left( {{1 \over 2},0,{{ - 1} \over 2}} \right)\]
\[ \Rightarrow PQ = {{\sqrt {34} } \over 2}\]