The exterior algebra by itself is not enough for physics, since it doesn’t “know” about the inner product, and so it cannot tell when vectors or wedges should b

The exterior algebra by itself is not enough for physics, since it doesn’t “know” about the inner product, and so it cannot tell when vectors or wedges should be perpendicular. Given the inner product, you can define the “hodge star”, which restores this information. Another way of restoring the information is to use a modification of the exterior algebra, where rather than having the product uv = u wedge v, we deform the product by setting uv = u wedge v + <u, v>. This is called a Clifford algebra [1], and when the input vector space is R^3 and the input inner product is the Euclidean dot product, this Clifford algebras is exactly the “geometric algebra”. I prefer to read about Clifford algebras, because the language is much less sensationalistic than many of the articles I’ve seen on geometric algebra. It also generalises to not just inner products, but any symmetric bilinear form, such as the Minkowski form on R^4 which is used in the study of relativity. [1] https://en.wikipedia.org/wiki/Clifford_algebra https://en.wikipedia.org/wiki/Clifford_algebra