4 A surface \(S\) has equation \(\mathrm { g } ( x , y , z ) = 0\), where \(\mathrm { g } ( x , y , z ) = ( y - 2 x ) ( y + z ) ^ { 2 } - 18\).
- Show that \(\frac { \partial \mathrm { g } } { \partial y } = ( y + z ) ( - 4 x + 3 y + z )\).
- Show that \(\frac { \partial \mathrm { g } } { \partial x } + 2 \frac { \partial \mathrm {~g} } { \partial y } - 2 \frac { \partial \mathrm {~g} } { \partial \mathrm { z } } = 0\).
- Hence identify a vector which lies in the tangent plane of every point on \(S\), explaining your reasoning.
- Find the cartesian equation of the tangent plane to the surface \(S\) at the point \(\mathrm { P } ( 1,4 , - 7 )\).
The tangent plane to the surface \(S\) at the point \(\mathrm { Q } ( 0,2,1 )\) has equation \(6 x - 7 y - 4 z = - 18\).
- Find a vector equation for the line of intersection of the tangent planes at P and Q .