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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.

Haider
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Aviv Cohn
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1 Answers1

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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.

Rich
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