Magnetic Field, Proton, Dynamics

[Dekoy]

Homework Statement

A proton of charge +e and mass mp enters a uniform magnetic field B =B\hat{i}with an initial velocity Vi = vix\hat{i}+ viy\hat{j}.
Without assuming any circular motion, show that its veloc-
ity v at any later time t is given by

v(t) = vix\hat{i}+ viy cos(eBt/mp)\hat{j}-viy sin(eBt/mp)\hat{k}

Homework Equations

F=qvB
(mv^2)/r=F
F=ma

The Attempt at a Solution

I have no idea where to go with this problem I drew a diagram for the proton in the magnetic field but i don't see the reason for the z component on the final velocity. I tried finding the time through a=v/t and go from there but couldn't get anything.
Thanks

 

[xcrunner2414]

I have no idea where to go with this problem I drew a diagram for the proton in the magnetic field but i don't see the reason for the z component on the final velocity. I tried finding the time through a=v/t and go from there but couldn't get anything.
Thanks

- The reason for the z-component of the final velocity is because there is a y-component to the initial velocity. The magnetic field is in the x-direction, and \hat j \times \hat i = -\hat k