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:

  ${\displaystyle a \wedge b .}$

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

  ${\displaystyle a \wedge b \wedge c. }$

• In a 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]

References

This page uses content from Wikipedia. The original article was at Blade (geometry). The list of authors can be seen in the page history. As with the Math Wiki, the text of Wikipedia is available under the Creative Commons Licence.