Official solution
Family of Plane's equation can be given by
\[(5 + 8\lambda )x + (8 - 7\lambda )y + (13 + \lambda )z - (29 + 20\lambda ) = 0\]
P1 passes through (2, 1, 3)
\[ \Rightarrow (10 + 16\lambda ) + (8 - 7\lambda ) + (39 + 3\lambda ) - (29 + 20\lambda ) = 0\]
\[ \Rightarrow - 8\lambda + 28 = 0 \Rightarrow \lambda = {7 \over 2}\]
d.r, s of normal to P1
\[\left\langle {33,{{ - 33} \over 2},{{33} \over 2}} \right\rangle \] or \[\left\langle {1, - {1 \over 2},{1 \over 2}} \right\rangle \]
P2 passes through (0, 1, 2)
\[ \Rightarrow 8 - 7\lambda + 26 + 2\lambda - (29 + 20\lambda ) = 0\]
\[ \Rightarrow 5 - 25\lambda = 0\]
\[ \Rightarrow \lambda = {1 \over 5}\]
d.r, s of normal to P2
\[\left\langle {{{33} \over 5},{{33} \over 5},{{66} \over 5}} \right\rangle \] or \[\left\langle {1,1,2} \right\rangle \]
Angle between normals
\[ = {{\left( {\widehat i - {1 \over 2}\widehat j + {1 \over 2}\widehat k} \right).\,\left( {\widehat i + \widehat j + 2\widehat k} \right)} \over {{{\sqrt 3 } \over 2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sqrt 6 }}\]
\[\cos \theta = {{1 - {1 \over 2} + 1} \over 3} = {1 \over 2}\]
\[\theta = {\pi \over 3}\]