Blade (geometry)

In the study of geometric algebras, a k-blade or a simple k-vector is a generalization of the concept of scalars and vectors to include simple bivectors, trivectors, etc. Specifically, a k-blade is a k-vector that can be expressed as the exterior product (informally wedge product) of 1-vectors, and is of grade k. 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 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: In a vector space of dimension n, a blade of grade n − 1 is called a pseudovector or an antivector. The highest grade element in a space is called a pseudoscalar, and in a space of dimension n is an n-blade. In a vector space of dimension n, there are k(n − k) + 1 dimensions of freedom in choosing a k-blade for 0 ≤ k ≤ n, of which one dimension is an overall scaling multiplier. 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. Indeed, a k-blade is naturally equivalent to a k-subspace endowed with a volume form (an alternating k-multilinear scalar-valued function) normalized to take unit value on the k-blade. 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 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.