Geometric algebra (GA) is a structure you add on top of a regular vector space. It was repopularized by David Hestenes in the 60s and 70s as a different method

Geometric algebra (GA) is a structure you add on top of a regular vector space. It was repopularized by David Hestenes in the 60s and 70s as a different method for describing spacetime (for physics). The basic observation of GA is this: in R^3 we've got scalars and vectors — but a vector is really a 1D subspace of R^3; so, the question is: why aren't there 0-, 2-, and 3- dimensional subspaces of the "usual" R^3 vector space 'in the algebra'? The answer to that question is a new thing called a Clifford algebra (the basis of GA), which we write like "CL(R^3)". David Hestenes noticed that CL(R^3) has a lot of physically-useful properties (although, he used CL(R^3;3,1) — the "3,1" describe the underlying metric/inner-product of the space; and sometimes, a conformal 5D space, instead). The basic object of GA is a "multivector": it is a "polynomial" of different subspaces: `A + BV + CV^V + DV^V^V ... ZV^...^V`. GA defines a special "multiplication" of the "polynomials" which is called the geometric product. The geometric product has a bunch of useful properties. However, as opposed to what the a sibling comment implies, the dimensionality of Clifford algebras is 2^N for an N-dimensional carrier. Thus, CL(R^3) is eight dimensional. The "nice" 5D conformal Clifford algebra is 32 dimensional. The inner-product is a square matrix with sides equal to the dimensionality of the Clifford algebra. Thus, what you might call a "dot product" in R^3 is an 8x8 matrix multiply in CL(R^3); for physically interesting algebras, it is a 32x32 matrix multiply.