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

[Swapnil]

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

 

[Hurkyl]

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.

 

[Swapnil]

So, if
$$\vec{C} = \vec{A}\times\vec{B}$$

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?

 

[Hurkyl]

Almost: $A \times B$ is the (pseudo)vector that is dual to the bivector $A \wedge B$.

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

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