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Angle Between Two Planes
Let 
and 
be two planes
in the three-dimensional Euclidean space
. The angle
between these planes is defined by means of the normal vectors
and
of and through the relationship
and
and the denominator
is product
of the lengths
of
and
. The formula implies
that the angle 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.
Generalization. The above definition can be generalized, at least locally, to a pair of intersecting differentiable
surfaces
in
. Given two differentiable surfaces 
and 
and a point
, the angle between and at 
is defined to be the angle between the tangent planes
and 
.