normal line and tangent plane via the gradient

The gradient $\nabla F$ of an implicit surface $F(x,y,z)=k$ is a ready-made normal vector, letting the tangent plane be written down without solving for $z$ .

A normal vector plus a point is all a plane needs. The gradient $\nabla F$ of a level surface $F(x,y,z)=k$ is exactly that normal vector at any point on the surface, so the tangent plane to an implicitly-defined surface is the point-normal plane through that point with normal $\nabla F$ , and the normal line is $\langle x,y,z\rangle=\langle x_0,y_0,z_0\rangle+t\,\nabla F(x_0,y_0,z_0)$ . The special case $F(x,y,z)=f(x,y)-z$ recovers the familiar tangent-plane formula from $\langle f_x,f_y,-1\rangle$ exactly; its real payoff is on surfaces where isolating $z$ isn't possible at all.

Examples

For $x^2+y^2-z^2=1$ at $(1,1,1)$ , $\nabla F(1,1,1)=\langle2,2,-2\rangle$ gives tangent plane $x+y-z=1$ and normal line $\langle1,1,1\rangle+t\langle1,1,-1\rangle$ .