PlanetMath: angle between two planes

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angle between two planes

(Definition)

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

$\displaystyle \cos\theta = \Big\vert \frac{\langle \boldsymbol{n}_1,\boldsymbol... ...2\rangle }{\Vert \boldsymbol{n}_1 \Vert \Vert \boldsymbol{n}_2 \Vert}\Big\vert,$

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

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

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:** Angle between two planes
$\includegraphics{angle.eps}$

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 and and a point $p\in S_1\cap S_2$ , the angle between and at is defined to be the angle between the tangent planes and .