About an example related to angular momentum in a textbook

[conectado]

Hey I either have the worst physics textbook in the history of physics textbooks or i can't understand a topic.I'd appreciate if you help me decide which is it.
(Pic related is the image of the example)
First let's agree on the symbols let's call angular velocity: $\omega = \frac{rxv}{||r||^2}$ , angular momentum $L=rxmv$ and Lineal Momentum:$P=mv$. Being that defined, the book attemps to proof that $L$ and \(\displaystyle \omega\) have generally different directions, here is the first problem i encountered, for what i know $m||r||^2\omega = L$ being $m||r||^2$ That means that Angular Velocity and Angular Momentum are parallel therefore have the same direction, the example proceed as following: 2 puntual mass are united by a rigid bar of despicable mass, therefore both have the same angular momentum. the momentum of body 1 is $L_1 = r_1xm_1v_1$ and it explicitly say that it's easy to see that it's contained in the plane given by $P_1 P_2 \omega$ (which i fail to see) being the direction perpendiculat to the line that unites $P_1$ with $P_2$ and that $L_2$ Has the same direction that $L_1$. Therefore the Angular Impulse make an angle of $\frac{\pi}{2}-\alpha$ with the axis of rotation($\omega$). Well that'd be an example i'd really appreciate if someone can clarify my doubts, i don't have a teacher to ask since i study by my own.
Here is a picture of the example:

 

[alena_S]

It looks like your textbook is trying to explain why the angular momentum and angular velocity vectors are usually not in the same direction. The angular momentum vector is a vector that points from the point mass to the axis of rotation, and its magnitude is the product of the mass and the distance from the axis of rotation times the velocity. The angular velocity vector points along the axis of rotation, and its magnitude is the product of the radius and the velocity.In the example, they explain how if two point masses are connected by a rigid bar, then the angular momentum vector of each point mass will be in the same direction, but the angular velocity vector will not be in the same direction. This is because the angular velocity vector is always perpendicular to the line connecting the two point masses, and the angular momentum vector is always in the plane defined by the two point masses and the angular velocity vector.This example should make it clear why the angular momentum and angular velocity vectors usually do not have the same direction. Hope this helps!