I am trying to find $\vec{r}(t)$ starting from $ \sum \vec{F} = m\vec{a} $ where $\vec{r}(t)$ is the flight of a cricket ball. The problem I have is that none of the forces (Magnus effect) that acts on a cricket ball and causes it to drift works perpendicular to the spin axis and the direction of travel.
The formula I am using for this is: $$\vec{a}(t) = \frac{16}3 \pi^2 c r^3 \rho \omega v \left\lVert \vec{u}(t) \times \vec{s} \right\rVert$$ Where $ \frac{16}3 \pi^2 c r^3 \rho \omega v$ are a whole bunch of constants (lift coefficient, radius, air pressure, spin, wind velocity respectively).
$\vec{u}(t)$ is pthe direction of motion, $\vec{s}$ is the spin axis (which is orthogonal to the seam of the cricket ball). The force operates at right angles to both of these vectors,
I am unsure how to integrate this in order to handle the cross product as obviously $\vec{r}(t)$ travels in its own direction. Any advice would be greatly appreciated.