Why?
My particle is moving along z and v=[0,0,v
z]
of course it will have a displacement along x and y but the intial velocity has only one component.
Could you please explain your point?
 Quote by tiny-tim
Where do the 1/v come from? 
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it comes from the differential equation:
dv
x/dt=qE
x/m
dv
y/dt=qB
xv
z/m
I change the variable t in z... it is a bit of algebra I can write you everything if you want so you can double check my manipulation.