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## Summary

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:[1]

• 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:
${\displaystyle a\wedge b.}$
• 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 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(nk) + 1 dimensions of freedom in choosing a k-blade for 0 ≤ kn, of which one dimension is an overall scaling multiplier.[5]

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] Indeed, a k-blade is naturally equivalent to a k-subspace, up to a scalar factor. When the space is endowed with a volume form (an alternating k-multilinear scalar-valued function), such a k-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.

## Notes

1. ^ Marcos A. Rodrigues (2000). "§1.2 Geometric algebra: an outline". Invariants for pattern recognition and classification. World Scientific. p. 3 ff. ISBN 981-02-4278-6.
2. ^ William E Baylis (2004). "§4.2.3 Higher-grade multivectors in Cℓn: Duals". Lectures on Clifford (geometric) algebras and applications. Birkhäuser. p. 100. ISBN 0-8176-3257-3.
3. ^ Lengyel, Eric (2016). Foundations of Game Engine Development, Volume 1: Mathematics. Terathon Software LLC. ISBN 978-0-9858117-4-7.
4. ^ John A. Vince (2008). Geometric algebra for computer graphics. Springer. p. 85. ISBN 978-1-84628-996-5.
5. ^ For Grassmannians (including the result about dimension) a good book is: Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library, New York: John Wiley & Sons, ISBN 978-0-471-05059-9, MR 1288523. The proof of the dimensionality is actually straightforward. Take the exterior product of k vectors ${\displaystyle v_{1}\wedge \cdots \wedge v_{k}}$  and perform elementary column operations on these (factoring the pivots out) until the top k × k block are elementary basis vectors of ${\displaystyle \mathbb {F} ^{k}}$ . The wedge product is then parametrized by the product of the pivots and the lower k × (nk) block. Compare also with the dimension of a Grassmannian, k(nk), in which the scalar multiplier is eliminated.
6. ^ David Hestenes (1999). New foundations for classical mechanics: Fundamental Theories of Physics. Springer. p. 54. ISBN 0-7923-5302-1.

## References

• David Hestenes; Garret Sobczyk (1987). "Chapter 1: Geometric algebra". Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics. Springer. p. 1 ff. ISBN 90-277-2561-6.
• Chris Doran & Anthony Lasenby (2003). Geometric algebra for physicists. Cambridge University Press. ISBN 0-521-48022-1.
• A Lasenby, J Lasenby & R Wareham (2004) A covariant approach to geometry using geometric algebra Technical Report. University of Cambridge Department of Engineering, Cambridge, UK.
• R Wareham; J Cameron & J Lasenby (2005). "Applications of conformal geometric algebra to computer vision and graphics". In Hongbo Li; Peter J Olver & Gerald Sommer (eds.). Computer algebra and geometric algebra with applications. Springer. p. 329 ff. ISBN 3-540-26296-2.