angle between two planes

Let $\pi_{1}$ and $\pi_{2}$ be two planes in the three-dimensional Euclidean space $\mathbb{R}^{3}$.  The angle $\theta$ between these planes is defined by means of the normal vectors $\boldsymbol{n}_{1}$ and $\boldsymbol{n}_{2}$ of $\pi_{1}$ and $\pi_{2}$ through the relationship

 $\cos\theta=\Big{|}\frac{\langle\boldsymbol{n}_{1},\boldsymbol{n}_{2}\rangle}{% \|\boldsymbol{n}_{1}\|\|\boldsymbol{n}_{2}\|}\Big{|},$

where the numerator is the inner product of $\boldsymbol{n}_{1}$ and $\boldsymbol{n}_{2}$ and the denominator is product of the lengths of $\boldsymbol{n}_{1}$ and $\boldsymbol{n}_{2}$.  The formula implies that the angle $\theta$ satisfies

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

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.

Generalization.  The above definition can be generalized, at least locally, to a pair of intersecting differentiable surfaces in $\mathbb{R}^{3}$.  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 $T_{p}(S_{2})$.

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