The vector given by the vector field at any point (x, y, z) is tangent to the flow line at that point. One thing I notice immediately is that that k component of the vector field is 0! That simplifies things a lot! It means that the flow is alway over a constant z and reduces this to a two dimensional problem- we only need to find x and y.
And in the xy-plane, if the vector fields is given by <A, B> that means that dx/dt is a multiple of A and that dy/dt is that same multiple of B. And that means that .
The result of all that, for this problem, is that .
That's a "separable" first order differential equation which should be easy to solve.