Prove that if the normals of two hyperplanes are orthogonal, the vectors on the hyperplanes are orthogonal

Question

For two planes:

$(\vec{p} - \vec{p_0})^T \vec{n_p} = 0$

$(\vec{q} - \vec{q_0})^T \vec{n_q} = 0$

Prove if the normals are orthogonal, i.e. $\vec{n_p}^T \vec{n_0} = 0$, then all vectors (except potentially the vectors along the intersection of two planes) $\vec{p}-\vec{p_0}$ and $\vec{q} - \vec{q_0}$ are orthogonal.

Answer 1

Isn't this not true? Consider the hyperplanes in $\mathbb R^3$ with normals $(1, 0, 0)$ and $(0, 1, 0)$, respectively. These normals are orthogonal, but both planes contain the vector $(0, 0, 1)$.