Geometric algebra
![Given two vectors a and b, if the geometric product ab is[6] anticommutative; they are perpendicular (top) because a ∧ b = −(b ∧ a) and a · b = 0, if it is commutative; they are parallel (bottom) because a ∧ b = 0 and a · b = b · a.](/Images/godic/202501/15/GA_parallel_and_perpendicular_vectors.svg2832.png")
![Orientation defined by an ordered set of vectors.Reversed orientation corresponds to negating the exterior product.Geometric interpretation of grade n elements in a real exterior algebra for n = 0 (signed point), 1 (directed line segment, or vector), 2 (oriented plane element), 3 (oriented volume). The exterior product of n vectors can be visualized as any n-dimensional shape (e.g. n-parallelotope, n-ellipsoid); with magnitude (hypervolume), and orientation defined by that on its n − 1-dimensional boundary and on which side the interior is.[7][8]](/Images/godic/202501/15/N_vector_positive_png_version2832.png")
![](/Images/godic/202501/15/N_vector_negative_png_version2832.png")
![In 3d space, a bivector a∧b defines a 2d plane subspace (light blue, extends infinitely in indicated directions). Any vector c in the 3-space can be projected onto and rejected normal to the plane, shown respectively by c⊥ and c∥.](/Images/godic/202501/15/GA_plane_subspace_and_projection.svg2832.png")
A geometric algebra (GA) is a Clifford algebra of a vector space over the field of real numbers endowed with a quadratic form. The term is also sometimes used as a collective term for the approach to classical, computational and relativistic geometry that applies these algebras. The Clifford multiplication that defines the GA as a unital ring is called the geometric product. Taking the geometric product among vectors can yield bivectors, trivectors, or general n-vectors. The addition operation combines these into general multivectors, which are the elements of the ring. This includes, among other possibilities, a well-defined formal sum of a scalar and a vector.