Blade (geometry)

{{Short description|Exterior product of vectors}}

{{About||the physical properties of a sword blade|Blade geometry}}

In the study of geometric algebras, a {{math|k}}-blade or a simple {{math|k}}-vector is a generalization of the concept of scalars and vectors to include simple bivectors, trivectors, etc. Specifically, a {{math|k}}-blade is a Multivector that can be expressed as the exterior product (informally wedge product) of 1-vectors, and is of grade {{math|k}}.

In detail:{{cite book |title=Invariants for pattern recognition and classification |author=Marcos A. Rodrigues |chapter=§1.2 Geometric algebra: an outline |chapter-url=https://books.google.com/books?id=QbFSt0SlDjIC&pg=PA3 |page=3 ff |isbn=981-02-4278-6 |year=2000 |publisher=World Scientific}}

• 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 {{math|a}} and {{math|b}}:
• : $a\; \backslash wedge\; b\; .$
• A 3-blade is a simple trivector, that is, it may be expressed as the wedge product of three vectors {{math|a}}, {{math|b}}, and {{math|c}}:
• : $a\; \backslash wedge\; b\; \backslash wedge\; c.$
• In a vector space of dimension {{math|n}}, a blade of grade {{math|n − 1}} is called a pseudovector{{cite book |chapter-url=https://books.google.com/books?id=oaoLbMS3ErwC&dq=%22pseudovectors+%28grade+n+-+1+elements%29%22&pg=PA100 |page=100 |author=William E Baylis |title=Lectures on Clifford (geometric) algebras and applications |isbn=0-8176-3257-3 |year=2004 |chapter=§4.2.3 Higher-grade multivectors in Cℓ_n: Duals |publisher=Birkhäuser}} or an antivector.{{cite book |title=Foundations of Game Engine Development, Volume 1: Mathematics |last1=Lengyel |first1=Eric |publisher= Terathon Software LLC|year=2016 |isbn=978-0-9858117-4-7}}
• The highest grade element in a space is called a pseudoscalar, and in a space of dimension {{math|n}} is an {{math|n}}-blade.{{cite book |url=https://books.google.com/books?id=3VxZqfm3I_MC&dq=pseudoscalar+%22highest+grade%22&pg=PA85 |title=Geometric algebra for computer graphics |author=John A. Vince |page=85 |isbn=978-1-84628-996-5 |year=2008 |publisher=Springer}}
• In a vector space of dimension {{math|n}}, there are {{math|1=k(n − k) + 1}} dimensions of freedom in choosing a {{math|k}}-blade for {{math|0 ≤ k ≤ n}}, of which one dimension is an overall scaling multiplier.For Grassmannians (including the result about dimension) a good book is: {{Citation | last1=Griffiths | first1=Phillip | author1-link=Phillip Griffiths | last2=Harris | first2=Joseph | author2-link=Joe Harris (mathematician) | title=Principles of algebraic geometry | publisher=John Wiley & Sons | location=New York | series=Wiley Classics Library | isbn=978-0-471-05059-9 | mr=1288523 | year=1994}}. The proof of the dimensionality is actually straightforward. Take the exterior product of {{math|k}} vectors $v\_1\backslash wedge\backslash cdots\backslash wedge\; v\_k$ and perform elementary column operations on these (factoring the pivots out) until the top {{math|k × k}} block are elementary basis vectors of $\backslash mathbb\{F\}^k$. The wedge product is then parametrized by the product of the pivots and the lower {{math|1=k × (n − k)}} block. Compare also with the dimension of a Grassmannian, {{math|k(n − k)}}, in which the scalar multiplier is eliminated.

A vector subspace of finite dimension {{math|k}} may be represented by the {{math|k}}-blade formed as a wedge product of all the elements of a basis for that subspace.{{cite book |title=New foundations for classical mechanics: Fundamental Theories of Physics | author=David Hestenes |url=https://books.google.com/books?id=AlvTCEzSI5wC&pg=PA54 | page=54 | isbn=0-7923-5302-1 | year=1999 | publisher=Springer}} Indeed, a {{math|k}}-blade is naturally equivalent to a {{math|k}}-subspace, up to a scalar factor. When the space is endowed with a volume form (an alternating {{math|k}}-multilinear scalar-valued function), such a {{math|k}}-blade may be normalized to take unit value, making the correspondence unique up to a sign.