In mathematics the geometric product is the product of Clifford algebra or geometric algebra. Either algebra is an algebra over a field, constructed from a vector space of a given dimension with the geometric product. As such it is fully specified by the geometric product over the vectors, so the properties of the product determine the properties of the algebra.

It is closely related to the inner product and exterior product, and so to the common vector dot and cross products. Like these is bilinear and distributive, and like the exterior product it is associative. But it is neither commutative nor anticommutative, except in a few instances which have particular geometric interpretations. Most notably it also has a simple inverse, making it far easier to use as equations involving the geometric product can often be simply solved using the inverse.

As suggested by the name the product and the algebra can be interpreted geometrically. There is not a single interpretation: rather many of the uses of the product are geometric or have geometric applications. These include determining parallel (geometry)ism or perpendicularity, generating reflections and rotations, and most of the geometric properties of complex numbers and quaternions.