Sub Nil Planes
This notebook show how the tangent plane changes along each axis

{f,g,h}={0,0,0};
Print["Planes along the x-axis"]
Animate[u={1,0,2 g};
v={0,1,-2 f};
e=Cross[u,v];
{a,b,c}=e;
d=e.{f,g,h};
Plot3D[(-a/c)*x+(-b/c)*y+d/c,{x,-50,50},{y,-50,50},PlotRange->{-100,100},Mesh->{{{f, Thick}}, {{g,Thick}}}],  {f, -50, 50,1}]

Print["Planes along the y-axis"]
Animate[u={1,0,2 g};
v={0,1,-2 f};
e=Cross[u,v];
{a,b,c}=e;
d=e.{f,g,h};
Plot3D[(-a/c)*x+(-b/c)*y+d/c,{x,-50,50},{y,-50,50},PlotRange->{-100,100},Mesh->{{{f, Thick}}, {{g,Thick}}}], {g, -50,50,1}]

Print["Planes along the z-axis"]
{f,g,h}={0,0,0};
Animate[u={1,0,2 g};
v={0,1,-2 f};
e=Cross[u,v];
{a,b,c}=e;
d=e.{f,g,h};
Plot3D[(-a/c)*x+(-b/c)*y+d/c,{x,-50,50},{y,-50,50},PlotRange->{-100,100},Mesh->{{{f, Thick}}, {{g,Thick}}}], {h, -50,50,1}]