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In geometric algebra, a blade is a generalization of the notion of vectors and scalars that includes bivectors, trivectors, etc. A k-blade is a blade of grade k where the grade is the dimension of the subspace the blade represents (not to be confused with the dimensionality of the space of the blades of that grade). In d-dimensional spaces, there blades of grade zero through d.

For example, in 2-dimensional space scalars are described as 0-blades, vectors are 1-blades, and area elements are 2-blades known as pseudoscalars, in that they are one-dimensional objects distinct from regular scalars.

In three-dimensional space, 0-blades are again scalars and 1-blades are three-dimensional vectors, but in three-dimensions, areas have an orientation, so while 2-blades are area elements, they are oriented. 3-blades (trivectors) represent volume elements and in three-dimensional space, these are scalar-like – i.e., 3-blades in three-dimensions form a one-dimensional vector space.

[edit] See also

• Multivector

[edit] External links

• A Geometric Algebra Primer, especially for computer scientists.