This is one of those questions which has confused a lot of students like me and I know similar questions have been asked on Physics Stack exchange but I literally want to know what was the reason behind defining it towards the axis of rotation and not any other ? Why did scientists used this convention ?
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$\begingroup$ What alternatives are you actively conceiving? $\endgroup$– Cosmas ZachosCommented Nov 6, 2020 at 17:01
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2$\begingroup$ Angular velocity is not really a vector (it is a "pseudovector"), and its "direction" simply marks the plane in which the rotation takes place. By the usual convention in analytic geometry, a plane is characterized by its normal (and a point), which happens to give the axis direction in this case. It also comes out of the cross product (of $r$ and $v$), matches curls of vector fields, etc., in other words, it is convenient for writing coordinate-free formulas. $\endgroup$– ConifoldCommented Nov 6, 2020 at 22:41
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2 Answers
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Not really discussing history, but since you asked ...
(Bold is vector, normal is magnitude)
The position vector is $\mathbf r$, the velocity vector is $\mathbf v$, and $\mathbf \omega$ is the angular velocity vector.
We know that the angular velocity and the velocity are related through $v = R\omega$, and, if you look carefully, you will see that $R = r\sin\alpha$, hence $v = r\omega\sin\alpha$. Looks like the magnitude of a cross product!
This might very well give one the idea of defining an angular velocity vector such that $\mathbf v=\mathbf\omega\times\mathbf r$ or $\mathbf r\times\mathbf\omega$. Since the right hand rule is a thing, it seems best to define $\mathbf v=\mathbf\omega\times\mathbf r$, where $\mathbf\omega$ would be in the direction your thumb points at when you close your hands in the direction of rotation.
Why is this useful? We can use any theorem that we have found for the cross product. :)
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Angular velocity is not a vector and not is it a pseudo-vector. These are in fact abbreviations for the correct notion.
A rotation in 3d has an axis of rotation. However, when we look at rotations in higher dimensions, say for example in 4d or 5d, the notion of an axis of rotation does not generalise (we can find one in odd dimensions but not in even ones).
However, rotations in 3D occurs in an invariant plane and its this plane that generalises to rotations in higher dimensions.
Such planes are indicated by bivectors. This is what it sounds like - two chosen vectors which are not linearly independent and so they span a plane.
A bivector in 3d is equivalent to a pseudo-vector, which is where this latter notion comes in, in discussions of angular velocity or momentum and the like, for example - torque.
I'd also add, given some of the nonsense said both for and against higher dimensions, that here is an example of thinking in higher dimensions which is helpful in thinking about our own spatial 3d. But nonetheless, it says nothing about the actual existence of higher dimensions.
