I am working through the classical mechanics textbook SICM[1] and this question is specific to something in the book. I understand in general how conservation of angular momentum leads to orbital motion being confined to a plane in the two-body case. I am just trying to understand the specific way that the textbook justifies it.
I follow till the part where they show using some Lagrangian formalism that the $z$ component of the angular momentum is conserved. And then it says:
The choice of the z-axis is arbitrary, so the conservation of any component of the angular momentum implies the conservation of all components. Thus the total angular momentum is conserved. We can choose the $z$-axis so all of the angular momentum is in the z component. Since $\vec{x}\cdot(\vec{x} \times \vec{v}) = \vec{v}\cdot(\vec{x}\times\vec{x} )=0$, the motion is confined to the plane perpendicular to the angular momentum: $\theta = \pi/2$, and $\dot{\theta}=0$.
I know that angular momentum is defined as $L = \vec{r} \times (m\vec{v})$. How do you go from there to $\vec{x}\cdot(\vec{x} \times \vec{v})$?
The relevant section in the book is linked at [1].