The position of x will be determined by vx: (similar conditions for y,z)

x(t)=x(0)+vx *t (if velocity is a constant value).
x(t+dt)=x(t)+vx(t)*dt (if velocity is not a constant, and dt is a small time interval).
The complete integration can be calculated with numerical method. For example: I use Runge-Kutta 4th order method for all the simulation in this web site.