Pseudo and Real Vectors

Psedo Vectors are ones that commute with Patiry operation, namely

pi+ V pi=V

When we flip everything around a origin of a given coordinate, (using the active picture of rotations, where coordinates remain), those vectors are intact. 

Such vectors are L, B, usually ones that come out from a CURL, like L=rxp, B=CURL(A)

real vectors are real physical vectors, e.g. r, v, p (linear momentum), E (electrical field)

where 

pi+ V pi=-V

that's to say, when we flip the world, they flip their directions also.

One amazingly interesting thing is that, if we compress our world into 2 Dimension, what we got is:

3D Real Vectors-> 2D Real Vectors

3D Pseudo Vectors-> Scalars!

It's intriging that Peusovectors would become scalars. Perhaps the best way of imaging this is to think of a CURL operation matrix and remove the uppermost row, that contains all the vector bases, since it has to be 2 by 2 matrix and we can't remove anything from rows that have 2D coordinates. Then the result is ostensibly :

B=dAy/dx-dAx/dy,  a scalar

similarly

L=rxpy-rypx 

(Discussion reference: http://bohr.physics.berkeley.edu/classes/221/1011/hws/hw25.pdf)