Editing Pseudovector (section)

==Geometric algebra==
In [[geometric algebra]] the basic elements are vectors, and these are used to build a hierarchy of elements using the definitions of products in this algebra. In particular, the algebra builds pseudovectors from vectors.

The basic multiplication in the geometric algebra is the [[geometric product]], denoted by simply juxtaposing two vectors as in '''ab'''. This product is expressed as:

:<math> \mathbf {ab} = \mathbf {a \cdot b} +\mathbf {a \wedge b} \ , </math>

where the leading term is the customary vector [[dot product]] and the second term is called the [[exterior algebra|wedge product or exterior product]]. Using the postulates of the algebra, all combinations of dot and wedge products can be evaluated. A terminology to describe the various combinations is provided. For example, a [[Multivector#Geometric algebra|multivector]] is a summation of ''k''-fold wedge products of various ''k''-values. A ''k''-fold wedge product also is referred to as a [[blade (geometry)|''k''-blade]].

In the present context the ''pseudovector'' is one of these combinations. This term is attached to a different multivector depending upon the [[dimension]]s of the space (that is, the number of [[linearly independent]] vectors in the space). In three dimensions, the most general 2-blade or [[bivector]] can be expressed as the wedge product of two vectors and is a pseudovector.<ref name=Pezzaglia>
{{cite book |title=Deformations of mathematical structures II  |chapter-url=https://books.google.com/books?id=KfNgBHNUW_cC&pg=PA131 |page=131 ''ff'' |isbn=0-7923-2576-1 |author=William M Pezzaglia Jr.|editor=Julian Ławrynowicz |year=1992 |publisher =Springer |chapter=Clifford algebra derivation of the characteristic hypersurfaces of Maxwell's equations}}
</ref> In four dimensions, however, the pseudovectors are [[multivector|trivectors]].<ref name=DeSabbata>

In four dimensions, such as a [[Dirac algebra]], the pseudovectors are [[multivector|trivectors]]. {{cite book |title=Geometric algebra and applications to physics |author1=Venzo De Sabbata |author2=Bidyut Kumar Datta |url=https://books.google.com/books?id=AXTQXnws8E8C&q=bivector+trivector+pseudovector+%22geometric+algebra%22&pg=PA64 |isbn=978-1-58488-772-0 |year=2007 |page=64 |publisher=CRC Press}}

</ref> In general, it is a {{nowrap|(''n'' − 1)}}-blade, where ''n'' is the dimension of the space and algebra.<ref name=Baylis01>

{{cite book |chapter-url=https://books.google.com/books?id=oaoLbMS3ErwC&q=%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''<sub>n</sub>: Duals |publisher=Birkhäuser}}

</ref> An ''n''-dimensional space has ''n'' basis vectors and also ''n'' basis pseudovectors. Each basis pseudovector is formed from the outer (wedge) product of all but one of the ''n'' basis vectors. For instance, in four dimensions where the basis vectors are taken to be {'''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub>, '''e'''<sub>4</sub>}, the pseudovectors can be written as: {'''e'''<sub>234</sub>, '''e'''<sub>134</sub>, '''e'''<sub>124</sub>, '''e'''<sub>123</sub>}.

===Transformations in three dimensions===
The transformation properties of the pseudovector in three dimensions has been compared to that of the [[vector cross product]] by Baylis.<ref name=Baylis>

{{cite book |author=William E Baylis |title=Theoretical methods in the physical sciences: an introduction to problem solving using Maple V |url=https://archive.org/details/theoreticalmetho0000bayl |url-access=registration |page=[https://archive.org/details/theoreticalmetho0000bayl/page/234 234], see footnote |isbn=0-8176-3715-X |year=1994 |publisher=Birkhäuser}}

</ref> He says: "The terms ''axial vector'' and ''pseudovector'' are often treated as synonymous, but it is quite useful to be able to distinguish a bivector from its dual." To paraphrase Baylis: Given two polar vectors (that is, true vectors) '''a''' and '''b''' in three dimensions, the cross product composed from '''a''' and '''b''' is the vector normal to their plane given by {{nowrap|1='''c''' = '''a''' × '''b'''}}. Given a set of right-handed orthonormal [[basis vector]]s {{nowrap|{ '''e'''<sub>ℓ</sub> }<nowiki/>}}, the cross product is expressed in terms of its components as:

:<math>\mathbf {a} \times \mathbf{b} = \left(a^2b^3 - a^3b^2\right) \mathbf {e}_1 + \left(a^3b^1 - a^1b^3\right) \mathbf {e}_2 + \left(a^1b^2 - a^2b^1\right) \mathbf {e}_3 ,</math>

where superscripts label vector components. On the other hand, the plane of the two vectors is represented by the [[exterior product]] or wedge product, denoted by {{nowrap|'''a''' ∧ '''b'''}}. In this context of geometric algebra, this [[bivector]] is called a pseudovector, and is the ''[[Hodge dual]]'' of the cross product.<ref name=Li>

{{cite book |title=Computer algebra and geometric algebra with applications |page=330 |chapter-url=https://books.google.com/books?id=uxofVAQE3LoC&q=%22is+termed+the+dual+of+x%22&pg=PA330 |author1=R Wareham, J Cameron  |author2=J Lasenby |author2-link=Joan Lasenby |name-list-style=amp |chapter=Application of conformal geometric algebra in computer vision and graphics |isbn=3-540-26296-2 |year=2005 |publisher=Springer}}  In three dimensions, a dual may be ''right-handed'' or ''left-handed''; see {{cite book |title=Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry |author1=Leo Dorst |author2=Daniel Fontijne |author3=Stephen Mann |chapter-url=https://books.google.com/books?id=-1-zRTeCXwgC&pg=PA82 |page=82 |chapter=Figure 3.5: Duality of vectors and bivectors in 3-D |isbn=978-0-12-374942-0|year=2007 |publisher=Morgan Kaufmann |edition=2nd}}

</ref> The ''dual'' of '''e'''<sub>1</sub> is introduced as {{nowrap|'''e'''<sub>23</sub> ≡}} {{nowrap|'''e'''<sub>2</sub>'''e'''<sub>3</sub> {{=}}}} {{nowrap|'''e'''<sub>2</sub> ∧ '''e'''<sub>3</sub>}}, and so forth. That is, the dual of '''e'''<sub>1</sub> is the subspace perpendicular to '''e'''<sub>1</sub>, namely the subspace spanned by '''e'''<sub>2</sub> and '''e'''<sub>3</sub>. With this understanding,<ref name=Perwass>

{{cite book |title=Geometric Algebra with Applications in Engineering |author=Christian Perwass |chapter-url=https://books.google.com/books?id=8IOypFqEkPMC&pg=PA17 |page=17 |chapter=§1.5.2 General vectors |isbn=978-3-540-89067-6 |year=2009 |publisher=Springer}}

</ref>

:<math> \mathbf{a} \wedge \mathbf{b} = \left(a^2b^3 - a^3b^2\right) \mathbf {e}_{23} + \left(a^3b^1 - a^1b^3\right) \mathbf {e}_{31} + \left(a^1b^2 - a^2b^1\right) \mathbf {e}_{12} \ . </math>

For details, see ''{{section link|Hodge star operator|Three dimensions}}''. The cross product and wedge product are related by:

:<math>\mathbf {a} \ \wedge \ \mathbf{b}  = \mathit i \ \mathbf {a} \ \times \ \mathbf{b} \ ,</math>

where {{nowrap|''i'' {{=}} '''e'''<sub>1</sub> ∧ '''e'''<sub>2</sub> ∧ '''e'''<sub>3</sub>}} is called the ''[[Pseudoscalar (Clifford algebra)#Unit pseudoscalar|unit pseudoscalar]]''.<ref name=Hestenes>
{{cite book |title=New foundations for classical mechanics: Fundamental Theories of Physics  |isbn=0-7923-5302-1 |edition=2nd |year=1999 |publisher=Springer |chapter=The vector cross product |author-link = David Hestenes
|author=David Hestenes |chapter-url=https://books.google.com/books?id=AlvTCEzSI5wC&pg=PA60 |page=60 }}
</ref><ref name=Datta>

{{cite book |title=Geometric algebra and applications to physics |chapter=The pseudoscalar and imaginary unit |chapter-url=https://books.google.com/books?id=AXTQXnws8E8C&pg=PA53 |page=53 ''ff'' |author1=Venzo De Sabbata |author2=Bidyut Kumar Datta |isbn=978-1-58488-772-0 |publisher=CRC Press |year=2007}}

</ref> It has the property:<ref name=Sobczyk>

{{cite book |title=Geometric algebra with applications in science and engineering |author1=Eduardo Bayro Corrochano |author2=Garret Sobczyk |url=https://books.google.com/books?id=GVqz9-_fiLEC&pg=PA126 |page=126 |isbn=0-8176-4199-8 |publisher=Springer |year=2001}}

</ref>

:<math>\mathit{i}^2 = -1 \ . </math>

Using the above relations, it is seen that if the vectors '''a''' and '''b''' are inverted by changing the signs of their components while leaving the basis vectors fixed, both the pseudovector and the cross product are invariant. On the other hand, if the components are fixed and the basis vectors '''e'''<sub>ℓ</sub>  are inverted, then the pseudovector is invariant, but the cross product changes sign. This behavior of cross products is consistent with their definition as vector-like elements that change sign under transformation from a right-handed to a left-handed coordinate system, unlike polar vectors.

===Note on usage===
As an aside, it may be noted that not all authors in the field of geometric algebra use the term pseudovector, and some authors follow the terminology that does not distinguish between the pseudovector and the cross product.<ref name=Jancewicz>

For example, {{cite book |author=Bernard Jancewicz |title=Multivectors and Clifford algebra in electrodynamics |url=https://books.google.com/books?id=seFyL-UWoj4C&pg=PA11 |page=11 |isbn=9971-5-0290-9 |year=1988 |publisher=World Scientific}}

</ref> However, because the cross product does not generalize to other than three dimensions,<ref name=Tischchenko1>

{{cite book |title=Linearity and the mathematics of several variables |author1=Stephen A. Fulling |author2=Michael N. Sinyakov |author3=Sergei V. Tischchenko |page=340 |url=https://books.google.com/books?id=Eo3mcd_62DsC&pg=RA1-PA340 |isbn=981-02-4196-8 |publisher=World Scientific |year=2000}}

</ref>
the notion of pseudovector based upon the cross product also cannot be extended to a space of any other number of dimensions. The pseudovector as a {{nowrap|1=(''n'' – 1)}}-blade in an ''n''-dimensional space is not restricted in this way.

Another important note is that pseudovectors, despite their name, are "vectors" in the sense of being elements of a [[vector space]]. The idea that "a pseudovector is different from a vector" is only true with a different and more specific definition of the term "vector" as discussed above.