Why is the dihedral angle between two planes, the angle between their normal vectors?

Question

I'm having trouble understanding why the angle between two planes (with the edge being their line of intersection), also known as the dihedral angle, is defined as the angle between their normal vectors.

Consider this sideways view of two planes $\textbf P_1$ and $\textbf P_2$. The red arrows denote their normal vectors $\textbf n_1$ and $\textbf n_2$ respectively:

Clearly, the angle between the normal vectors (green angle) is not the same as the angle between the planes along their line of intersection (blue angle).

What am I understanding wrong in this definition of the dihedral angle?

Answer 1

Note that we can consider two angles between the planes and the normal vectors, the first one $\alpha \le \frac \pi 2$ and the second one $\beta \ge \frac \pi 2$.

In your picture you are considering the acute angle $\alpha$ for the planes and the obtuse angle $\beta$ for the normal vectors and we have $\alpha=\pi - \beta$.