I am at the moment writing a script for a video talking about angular vectors and how to find them in discussion about how a top stays upright.
Currently, I am relating equations like $\vec v = \vec \omega r$ and $\vec \tau=r \vec F \sin \theta$ with the cross product, and how they are representations of the base formula $a \times b= |\vec a||\vec b|\sin \theta \hat n$, as for the first $\vec v$ is the cross product of $\vec \omega$ and $r$, and for the second, $\vec \tau$ is the cross product of $r$ and $\vec F$. Since experimentally these physical equations are true, and since based on precession angular momentum follows torque, the vector locations of the anguar quantities based on the cross product have to be true.
However, in reviewing the script, the problem I note is that I use geometric justification to move tangential velocity and force to the axis of rotation to make the cross product more clear, and I did so because the geometry of the situation allows for the vector to be moved:
Yet, this is problematic conceptually, as it matters were velocity and force occur on a rotating object. By moving the vectors, the object with the velocity and radius vectors is no longer rotating but translating forward, and by moving the vectors, the object with the force and radius vectors is no longer torqueing but accelerating forward.
Yet, based on what works in the geometry of the situation, it seems as if the objects can do both, translate or rotate based on the magnitude of the cross product encompassing both vector locations as to where the object rotates and where it translates, both sides of the radius position vector, yet clearly in real life the position of the force/velocity vector matters in regards to where it acts from the axis of rotation, as to if the object translates or rotates; only one or the other can be done, but the geometry suggest they lead to the same effect.
And, it seems as if this type of justification of moving the vector is used elsewhere to discuss rotation and the cross product, like in this graphic here which also moves the force vector, an image used in this post:
But, this just begs the question still: how should I make sense of the fact that the geometry of the situation would allow for the force/velocity vector to move, yet this produces a completely different result in real life based on where it is? Is there something I am failing to consider, that I got wrong here?