Convert newton formula to acceleration formulas

I am trying to understand a problem from the book Solving Problems in Scientific Computing Using Maple and MATLAB. The problem is about trajectories of tennis balls.
The author gives two formulas $D_L(v)$ for drag force and $M_L(v)$ for magnus force:
$$D_L(v)=C_Dfrac{1}{2}frac{pi d^2}{4}rho v^2$$
$$M_L(v)=C_Mfrac{1}{2}frac{pi d^2}{4}rho v^2$$
with $C_D$ and $C_M$ drag coefficient and magnus coefficient respectively.
Later he gives the trajectory vector by:
$$mfrac{d^2vec{r}(t)}{dt^2}=-mvec{g}-D_Lfrac{vec{v}}{v}+M_Lfrac{vec{omega}}{omega}timesfrac{vec{v}}{v}.$$
So far so good. But then he derives from that the following two formulas:
$$ddot{x}=-C_Dalpha vdot{x}+eta,C_Malpha vdot{z}$$
$$ddot{z}=-g-C_Dalpha vdot{z}-eta,C_Malpha vdot{x}$$
where $v=sqrt{dot{x}^2+dot{z}^2}$, $alpha=(rhopi d^2)/(8m)$ and $eta=pm 1$ is the direction of rotation.

If I understand correctly, $dot{x}, dot{z}$ are the velocities and $ddot{x}, ddot{z}$ are the accelerations in $x$ and $z$ direction.
I do not understand how to get to these equations by plugging $D_L$ and $M_L$ in above newton formula. For example, how do I handle the cross product on the right to summarize the sum? How do I get two $alpha$‘s on the right in both equations?
And one minor thing: why $L$ in the index of $D_L$ and $M_L$? Is it for lift?

Any help is appreciated! Let me know, if I forgot an important variable or other information is missing.