Proving that orbital motion is confined to a plane perpendicular to angular momentum

Question

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].

Answer 1

Our choice of the $z$-axis was arbitrary, so we can choose other axes and we will find that angular momentum along those axes is also conserved. This implies that the total angular momentum is conserved. In particular, you can choose a coordinate system, and then show that angular momentum is conserved along the $x, y$ and $z$–axes. Now, orient your $z$-axis in the direction of the angular momentum. To see that the orbital motion is perpendicular to the angular momentum, use the dot product: $\vec x\cdot \vec L = \vec x \cdot (\vec x\times m\vec v) = m\vec v\cdot (\vec x\times \vec x) = 0$. Since $\vec L$ is constant, $\vec x$ lies in a plane whose normal is $\vec L$.