angle between two planes

Let π1 and π2 be two planes in the three-dimensional Euclidean space 3.  The angle θ between these planes is defined by means of the normal vectorsMathworldPlanetmath 𝒏1 and 𝒏2 of π1 and π2 through the relationship


where the numerator is the inner product of 𝒏1 and 𝒏2 and the denominator is product of the lengths of 𝒏1 and 𝒏2.  The formula implies that the angle θ satisfies


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 differentiableMathworldPlanetmathPlanetmath surfaces in 3.  Given two differentiable surfaces S1 and S2 and a point pS1S2, the angle between S1 and S2 at p is defined to be the angle between the tangent planesMathworldPlanetmath Tp(S1) and Tp(S2).

Title angle between two planes
Canonical name AngleBetweenTwoPlanes
Date of creation 2013-03-22 16:19:41
Last modified on 2013-03-22 16:19:41
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 9
Author CWoo (3771)
Entry type Definition
Classification msc 51N20
Synonym angle between planes
Related topic AngleBetweenTwoLines
Related topic AngleBetweenLineAndPlane
Defines angle between differentiable surfaces