What is the meaning of angle between two planes?

Question

Suppose there are two planes that have normal vectors $\overrightarrow{n_1}$ and $\overrightarrow{n_2}$. Let vector $\overrightarrow{n_2'}$ = -$\overrightarrow{n_2}$. So, $\overrightarrow{n_2'}$ is also the normal vector of plane 2. We can see that the angle between $\overrightarrow{n_1}$ and $\overrightarrow{n_2}$, and between $\overrightarrow{n_1}$ and $\overrightarrow{n_2'}$ is different. Some say that the angle between two planes is the angle between their normals. If there are two possible angles between their normals, which one is the correct angle between the planes?

Note: I have the picture to help you guys understand my question easier, but somehow I can't upload it. Hope you guys understand my question.

Answer 1

By convention, the angle between the two planes is the smaller of the two angles you mention.

Answer 2

To think of the angle between two planes, imagine we cut both planes with a third "cross section" plane that is perpendicular to both. Then in the cross section intersection is represented by two lines in the third plane. The angle between those two lines is the angle between the planes.

But the angle between two lines can be measured on different sides of the point where they intersect. You'll get either an acute angle or an obtuse angle. The two angles will be supplementary.

Changing $n_2$ to $n_2'$ gives you the supplementary angle to the one you had first.

As Arthur says, in the absence of a reason to do otherwise one usually follows the convention of stating the acute angle between two planes rather than the obtuse one. A "reason to do otherwise" would be if you're in a situation where the planes are oriented or have definite "outer" and "inner" sides. Then it can make perfect sense to say that, for example, the angle between two sides in a (non-convex) polyhedron is $120^\circ$ rather than $60^\circ$.

Answer 3

In 3D, the concept of orientation ("clockwise" or "counterclockwise") disappears. Thus the planes indeed have two normals, which define four angles.

By convention, the angle belonging to the first or second quadrant ($[0,\pi)$ radians) is considered.