Blade (geometry)

In the study of geometric algebras, a -blade or a simple -vector is a generalization of the concept of scalars and vectors to include simple bivectors, trivectors, etc. Specifically, a -blade is a -vector that can be expressed as the exterior product of 1-vectors, and is of grade.
In detail:

A 0-blade is a scalar.
A 1-blade is a vector. Every vector is simple.
A 2-blade is a simple bivector. Sums of 2-blades are also bivectors, but not always simple. A 2-blade may be expressed as the wedge product of two vectors and :
:
A 3-blade is a simple trivector, that is, it may be expressed as the wedge product of three vectors,, and :
:
In vector spaces of dimension ≤ 3, every k-vector is a blade. In dimension ≥ 4, there exist k-vectors that are not blades. This makes the distinction important, because in higher dimensions most k-vectors do not correspond to any subspace, which requires defining blades separately from k-vectors to identify the k-vectors that actually do.
In a vector space of dimension, a blade of grade is called a pseudovector or an antivector.
The highest grade element in a space is called a pseudoscalar, and in a space of dimension is an -blade.
In a vector space of dimension, there are dimensions of freedom in choosing a -blade for, of which one dimension is an overall scaling multiplier.

A vector subspace of finite dimension may be represented by the -blade formed as a wedge product of all the elements of a basis for that subspace. Indeed, a -blade is naturally equivalent to a -subspace, up to a scalar factor. When the space is endowed with a volume form, such a -blade may be normalized to take unit value, making the correspondence unique up to a sign.

Examples

In two-dimensional space, scalars are described as 0-blades, vectors are 1-blades, and area elements are 2-blades in this context known as pseudoscalars, in that they are elements of a one-dimensional space that is distinct from regular scalars.
In three-dimensional space, 0-blades are again scalars and 1-blades are three-dimensional vectors, while 2-blades are oriented area elements. In this case 3-blades are called pseudoscalars and represent three-dimensional volume elements, which form a one-dimensional vector space similar to scalars. Unlike scalars, 3-blades transform according to the Jacobian determinant of a change-of-coordinate function.