Hall effect - semiclassical approach

[armandowww]

I'm perplexed by semiclassical treatment of Hall effect.
I'm referring to Aschroft Mermin text. At page 233 is my doubt:
why is it necessary the introduction of w drift velocity vector and why does it take that form?

 

[Gokul43201]

Recall the trajectory of a free charged particle in crossed E and B fields (from Jackson, or some SR course) - it is a cycloid with a drift velocity along E X B.

You can derive this by transforming to the frame where E' = 0.