Particle in two dimensions of the velocity vector
v. In the simplest case of circular motion at radius , with position given by the angular displacement \phi(t) from the x-axis, the orbital angular velocity is the rate of change of angle with respect to time: {{tmath|1= \textstyle \omega = \frac{d\phi}{dt} }}. If \phi is measured in
radians, the arc-length from the positive x-axis around the circle to the particle is , and the linear velocity is {{tmath|1= \textstyle v(t) = \frac{d\ell}{dt} = r\omega(t) }}, so that {{tmath|1= \textstyle \omega = \frac{v}{r} }}. In the general case of a particle moving in the plane, the orbital angular velocity is the rate at which the position vector relative to a chosen origin "sweeps out" angle. The diagram shows the position vector \mathbf{r} from the origin O to a particle , with its
polar coordinates . (All variables are functions of time .) The particle has linear velocity splitting as {{tmath|1= \mathbf{v} = \mathbf{v}_\Vert+\mathbf{v}_\perp }}, with the radial component \mathbf{v}_\| parallel to the radius, and the cross-radial (or tangential) component \mathbf{v}_\perp perpendicular to the radius. When there is no radial component, the particle moves around the origin in a circle; but when there is no cross-radial component, it moves in a straight line from the origin. Since radial motion leaves the angle unchanged, only the cross-radial component of linear velocity contributes to angular velocity. The angular velocity
ω is the rate of change of angular position with respect to time, which can be computed from the cross-radial velocity as: \omega = \frac{d\phi}{dt} = \frac{v_\perp}{r}. Here the cross-radial speed v_\perp is the signed magnitude of {{tmath| \mathbf{v}_\perp }}, positive for counter-clockwise motion, negative for clockwise. Taking polar coordinates for the linear velocity \mathbf{v} gives magnitude v (linear speed) and angle \theta relative to the radius vector; in these terms, , so that \omega = \frac{v\sin(\theta)}{r}. These formulas may be derived doing {{tmath|1= \mathbf{r}=(r\cos(\varphi),r\sin(\varphi)) }}, being r a function of the distance to the origin with respect to time, and \varphi a function of the angle between the vector and the x-axis. Then: \frac{d\mathbf{r}}{dt} = (\dot{r}\cos(\varphi) - r\dot{\varphi}\sin(\varphi), \dot{r}\sin(\varphi) + r\dot{\varphi}\cos(\varphi)), which is equal to: \dot{r}(\cos(\varphi), \sin(\varphi)) + r\dot{\varphi}(-\sin(\varphi), \cos(\varphi)) = \dot{r}\hat{r} + r\dot{\varphi}\hat{\varphi} (see
Unit vector in cylindrical coordinates). Knowing {{tmath|1= \textstyle \frac{d\mathbf{r} }{dt} = \mathbf{v} }}, we conclude that the radial component of the velocity is given by {{tmath| \dot{r} }}, because \hat{r} is a radial unit vector; and the perpendicular component is given by r\dot{\varphi} because \hat{\varphi} is a perpendicular unit vector. In two dimensions, angular velocity is a number with plus or minus sign indicating orientation, but not pointing in a direction. The sign is conventionally taken to be positive if the radius vector turns counter-clockwise, and negative if clockwise. Angular velocity then may be termed a
pseudoscalar, a numerical quantity which changes sign under a
parity inversion, such as inverting one axis or switching the two axes.
Particle in three dimensions 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 \mathbf{u} 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 {{tmath| \mathbf{u} }}). Taking polar coordinates (r,\phi) in this plane, as in the two-dimensional case above, one may define the orbital angular velocity vector as: : \boldsymbol\omega =\omega \mathbf u = \frac{d\phi}{dt}\mathbf u=\frac{v \sin(\theta)}{r}\mathbf u, where is the angle between {{tmath| \mathbf{r} }} and {{tmath| \mathbf{v} }}. In terms of the cross product, this is: : \boldsymbol\omega=\frac{\mathbf r\times\mathbf v}{r^2}. From the above equation, one can recover the tangential velocity as: : \mathbf{v}_{\perp} =\boldsymbol{\omega} \times\mathbf{r} == Spin angular velocity of a rigid body or reference frame ==