Blade (geometry)

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In the study of geometric algebras, a blade is a generalization of the concept of scalars
and vectors to include simple bivectors, trivectors, etc.

Specifically, a k-blade is any object that can be expressed as the exterior product (informally wedge product) of k vectors, and is of grade k.

In detail:[1]

A 0-blade is a scalar.
A 1-blade is a vector. Every vector is simple.

A 2-blade is a simple bivector. Linear combinations of 2-blades also are bivectors,
but need not be simple, and are hence not necessarily 2-blades.
A 2-blade may be expressed as the wedge product of two vectors a and b:

A 3-blade is a simple trivector, that is, it may be expressed as the wedge product of
three vectors a, b, and c:

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In a vector space of dimension n, a blade of grade n − 1 is called a pseudovector[2] or an antivector.[3]
The highest grade element in a space is called a pseudoscalar, and in a space of dimension n is an n-blade.[4]
In a vector space of dimension n, there are k(n − k) + 1 dimensions of freedom in choosing a k-blade,
of which one dimension is an overall scaling multiplier.[5]

For an n-dimensional space, there are blades of all grades from 0 to n inclusive.
A vector subspace of finite dimension k may be represented by the k-blade formed as
a wedge product of all the elements of a basis for that subspace.[6]