Equation of plane with zero vector as normal

Question

I'm wondering what's the equation of a plane with the zero vector as normal. Is it all planes: Ax+By+Cz=D? Thanks in advance!

Answer 1

In affine geometry, using the homogeneous coordinates $x=x_1/x_0, \; y=x_2/x_0, \; z=x_3/x_0$, a plane $Ax+By+Cz=D$ is written as $Ax_1 +Bx_2+ Cx_3-Dx_0=0$.

Putting $0=A=B=C$ you get $-Dx_0=0 \; \to x_0=0$ and that's the plane at "infinite".

Answer 2

In ordinary vector geometry, the set of elements normal to the zero vector do not determine a plane: all vectors are normal to $(0,0,0)$, so the set of vectors "normal/orthogonal" to zero is the entire space.

A (hyper)plane has dimension one less than the entire space, and you need a nonzero vector to determine a (hyper)plane using the ordinary dot product in $\mathbb R^n$ (which is where I assume you are working.)