MathTheBeautiful Tensors vs Space-time Numbers 4: Vector Components
Channel: Doug Sweetser
Date: 2017-06-21

A vector written in component form has both a magnitude and a direction expressed as a basis vector. For a vector space with an orthonormal basis, the dot product of two basis vectors will be unity if they are the same basis vector, and zero otherwise. A vector space with a non-othonormal of course be different.

Space-time numbers have relationships built into the division algebra. The dot product of any two unitary space-time numbers necessarily leads to another unitary number. One good consequence is the rules for handling rotations is built right in.