Differences between an axial vector, a pseudo vector and a bivector?

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What is the difference between an axial vector, a psudo vector and a bivector?
 

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  • #2
Hurkyl
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Axial vector and pseudo-vector seem to mean the same thing, and are dual to bivectors.

A bivector is a "directed area". (similar to how a vector is a "directed length") In three dimensions, a directed area can be represented by its normal vector (i.e. it's "axis"): that's where pseudovectors come from.

Similarly, in three dimensions, a "directed volume" can be represented by a number: that's where pseudoscalars come from.
 
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  • #3
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So, if
[tex] \vec{C} = \vec{A}\times\vec{B}[/tex]

then C would be a psudo vector and it would also be a bivector since its magnitude is equal to the area of the parallogram spanned by A and B?
 
  • #4
Hurkyl
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Almost: [itex]A \times B[/itex] is the (pseudo)vector that is dual to the bivector [itex]A \wedge B[/itex].

[itex]A \times B[/itex] is merely the properly oriented vector that is perpendicular to the oriented parallelogram with sides A and B. Roughly speaking, [itex]A \wedge B[/itex] is that oriented parallelogram.

(But only roughly speaking -- the picture isn't quite that nice. For example, [itex](A + B) \wedge B = A \wedge B[/itex])
 

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