Editing Angular velocity (section)

=== Particle in three dimensions ===
[[Image:Angular velocity.svg|class=skin-invert-image|thumb|250px|The orbital angular velocity vector encodes the time rate of change of angular position, as well as the instantaneous plane of angular displacement. In this case (counter-clockwise circular motion) the vector points up.]]

In  [[three-dimensional space]], we again have the position vector '''r''' of a moving particle. Here, orbital angular velocity is a [[pseudovector]] whose magnitude is the rate at which '''r''' sweeps out angle (in radians per unit of time), and whose direction is perpendicular to the instantaneous plane in which '''r''' sweeps out angle (i.e. the plane spanned by '''r''' and '''v'''). However, as there are ''two'' directions perpendicular to any plane, an additional condition is necessary to uniquely specify the direction of the angular velocity; conventionally, the [[right-hand rule]] is used.

Let the pseudovector <math>\mathbf{u}</math> be the unit vector perpendicular to the plane spanned by '''r''' and '''v''', so that the right-hand rule is satisfied (i.e. the instantaneous direction of angular displacement is counter-clockwise looking from the top of <math>\mathbf{u}</math>). Taking polar coordinates <math>(r,\phi)</math> in this plane, as in the two-dimensional case above, one may define the orbital angular velocity vector as:

: <math>\boldsymbol\omega =\omega \mathbf u = \frac{d\phi}{dt}\mathbf u=\frac{v \sin(\theta)}{r}\mathbf u,</math>

where ''θ'' is the angle between '''r''' and '''v'''. In terms of the cross product, this is:

: <math>\boldsymbol\omega
=\frac{\mathbf r\times\mathbf v}{r^2}.</math><ref>{{cite book |last1=Singh |first1=Sunil K. |title=Angular Velocity |url=https://cnx.org/contents/MymQBhVV@175.14:51fg7QFb@14/Angular-velocity |via=OpenStax |publisher=Rice University |access-date=21 May 2021 |ref=1}}</ref>

From the above equation, one can recover the tangential velocity as:

:<math>\mathbf{v}_{\perp} =\boldsymbol{\omega} \times\mathbf{r}</math>