angle between two planes

Let ${\pi }_1$ and ${\pi }_2$ be two planes in the three-dimensional Euclidean space ${ℝ}^3$.  The angle $\theta$ between these planes is defined by means of the normal vectors ${𝒏}_1$ and ${𝒏}_2$ of ${\pi }_1$ and ${\pi }_2$ through the relationship

$$\cos \theta =\left|\frac{⟨{𝒏}_1,{𝒏}_2⟩}{\parallel {𝒏}_1\parallel \parallel {𝒏}_2\parallel }\right|,$$

where the numerator is the inner product of ${𝒏}_1$ and ${𝒏}_2$ and the denominator is product of the lengths of ${𝒏}_1$ and ${𝒏}_2$.  The formula implies that the angle $\theta$ satisfies

$$0\le \theta \le \frac{\pi }{2}.$$

The quotient in the formula remains unchanged as one multiplies the normal vectors by some non-zero real numbers, so that the cosine is independent of the lengths of the chosen vectors.  Therefore, there is no ambiguity in this definition.

Figure 1: Angle between two planes

Generalization.  The above definition can be generalized, at least locally, to a pair of intersecting differentiable surfaces in ${ℝ}^3$.  Given two differentiable surfaces $S_1$ and $S_2$ and a point $p\in S_1\cap S_2$, the angle between $S_1$ and $S_2$ at $p$ is defined to be the angle between the tangent planes $T_p(S_1)$ and $T_p(S_2)$.