Editing Pseudovector (section)

===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.