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I would like to validate my understanding of angular velocity as a vector.

Suppose we have a particle $P$ moving around in $3D$ space in some arbitrary way. At any given point in time, we would like to know its current angular velocity $\vec\omega$ around a specific point $O$ in space. $\vec\omega$ is a vector (with $3$ components).

In any given time, the particle can be said to be on some $2D$ plane called $S$.

$S$ is defined to have the following vectors lay on it: the vector from $O$ to $P$, and the current velocity vector $\vec v$ of $P$. There is exactly one such possible plane at any given time.

The axis of rotation around which the particle is rotating at a given point in time, is (one of the two) perpendicular vectors to $S$. We will call this vector $\vec N$.

Please answer the two following questions:

  1. Is my understanding so far accurate?

  2. If so, is it correct to say that the $\vec\omega$ vector is always parallel to the vector $\vec N$?

Please note that I have done reading online on this topic. Based on this reading, I would like to understand if my framing of this concept is accurate.

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  • $\begingroup$ It is accurate, just notice that neither O or N is defined if the particle is not accelerated. I think it is also useful to notice that S is the plane that contains both the velocity and the acceleration vectors. From these two vectors you can find O and N. $\endgroup$
    – Wolphram jonny
    Commented Mar 23 at 17:16
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    $\begingroup$ If we call the position vector $\vec{r}$, then $\vec\omega = \frac{\vec{r}\times\vec{v}}{r^2}$. I believe this agrees with your analysis. $\endgroup$
    – The Photon
    Commented Mar 23 at 17:17
  • $\begingroup$ I've written some posts regarding this topic, which you can see here and here. $\endgroup$
    – Maximal Ideal
    Commented Mar 23 at 17:32
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    $\begingroup$ See also: 2021 answer (by me) to a question titled: why is angular momentum a vector The thrust of that answer: a more robust representation of angular momentum is in the form of a bivector; a geometric entity that has two vectors as its elements. $\endgroup$
    – Cleonis
    Commented Mar 23 at 21:43

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Your reasoning is correct. The angular velocity has to be normal to the plane of rotation. If it, or any component of it, lay in the plane of rotation, that vector quantity would have to be changing constantly as the particle rotated about its axis and its position and velocity vectors changed. Therefore it must be parallel to any other normal vector of that plane.

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