Normal vector or normal line to surface

2 A surface has equation \(x ^ { 2 } - 4 x y + 3 y ^ { 2 } - 2 z ^ { 2 } - 63 = 0\).

1. Find a normal vector at the point \(( x , y , z )\) on the surface.
2. Find the equation of the tangent plane to the surface at the point \(\mathrm { Q } ( 17,4,1 )\).
3. The point \(( 17 + h , 4 + p , 1 - h )\), where \(h\) and \(p\) are small, is on the surface and is close to Q . Find an approximate expression for \(p\) in terms of \(h\).
4. Show that there is no point on the surface where the normal line is parallel to the \(z\)-axis.
5. Find the two values of \(k\) for which \(5 x - 6 y + 2 z = k\) is a tangent plane to the surface.

2 A surface \(S\) has equation \(\mathrm { g } ( x , y , z ) = 0\), where \(\mathrm { g } ( x , y , z ) = x ^ { 2 } + 3 y ^ { 2 } + 2 z ^ { 2 } + 2 y z + 6 x z - 4 x y - 24\). \(\mathrm { P } ( 2,6 , - 2 )\) is a point on the surface \(S\).

1. Find \(\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }\) and \(\frac { \partial \mathrm { g } } { \partial z }\).
2. Find the equation of the normal line to the surface \(S\) at the point P .
3. The point Q is on this normal line and close to P . At \(\mathrm { Q } , \mathrm { g } ( x , y , z ) = h\), where \(h\) is small. Find, in terms of \(h\), the approximate perpendicular distance from Q to the surface \(S\).
4. Find the coordinates of the two points on the surface at which the normal line is parallel to the \(y\)-axis.
5. Given that \(10 x - y + 2 z = 6\) is the equation of a tangent plane to the surface \(S\), find the coordinates of the point of contact.

4 A surface \(S\) has equation \(\mathrm { g } ( x , y , z ) = 0\), where \(\mathrm { g } ( x , y , z ) = ( y - 2 x ) ( y + z ) ^ { 2 } - 18\).

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