Vector algebra and exterior algebra
The current form of vector algebra in three dimensions comes to us from lecture notes by Josiah Willard Gibbs, published in 1901. The scalar product and vector product satisfy various useful identities with no obvious overall structure. In particular they do not generalize in an obvious way to dimensions greater than three.
This talk will describe a natural generalization of vector algebra for any number of dimensions. The setting is the exterior algebra of Grassmann, equipped with both an exterior product and an interior product. These correspond to the creation and annihilation operator for a fermi system. (They may also be regarded as specializations of Clifford algebra constructions.)
When this generalization is specialized to three dimensions, it is seen that all the usual vector identities fall into a systematic pattern and have natural geometric meanings.