# 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. Figure 1: Angle between two planes

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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