In generalized geometric algebras {Grassmann algebra}, the basis elements are the unit-magnitude dimensions, which can be any number and can be non-orthogonal. Elements are dimension linear combinations and have grade, magnitude, direction, and direction sense.
Operations are reflections. Elements add to make a new element. Elements multiply to make an element of one higher dimension (wedge product) {Grassmann product, algebra}. Parallel vectors are commutative. Perpendicular vectors are anti-commutative. Elements are associative for addition and multiplication. Grassmann algebra [1844 and 1862] is Clifford algebra in which two successive reflections cancel, rather than making rotation, and so there are no rotations and no need for metric or perpendicularity.
Mathematical Sciences>Geometry>Kinds>Geometric Algebra
3-Geometry-Kinds-Geometric Algebra
Outline of Knowledge Database Home Page
Description of Outline of Knowledge Database
Date Modified: 2022.0224