Refer to the fig...
The field is in +i direction so v0x will remain un affected...
so v(x) at any later time = v0x and trajectory (1) will be a straight line...
Now for the trajectory (2) due to the y component
= eB/m
as the path followed will be circular so the y component along y will vary simple harmonically...
so using the equation of SHM v=A
cos
t
vy = v0ycos
t
For trajectory (3) due to z component
= eB/m
again as the motion due to v0z is circular so the z component of velocity varies simple harmonically...
vz=v0zcos(
t +
/2) { Clearly from the figure the phase difference due to the trajectories (2) and (3) is pi/2 }
=>vz= - v0z sin
t
so the vector equation at any later time will be
v = v0x
+ v0ycos
t
- v0z sin
t 