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

cosθ=|𝒏1,𝒏2𝒏1𝒏2|,

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

0θπ2.

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