, the lowercase Greek letter omega), also known as the angular frequency vector,[1] is a pseudovector representation of how the angular position or orientation of an object changes with time, i.e. how quickly an object rotates (spins or revolves) around an axis of rotation and how fast the axis itself changes direction.

is normal to the instantaneous plane of rotation or angular displacement.

The radian is a dimensionless quantity, thus the SI units of angular velocity are dimensionally equivalent to reciprocal seconds, s−1, although rad/s is preferable to avoid confusion with rotation velocity in units of hertz (also equivalent to s−1).

[4] The sense of angular velocity is conventionally specified by the right-hand rule, implying clockwise rotations (as viewed on the plane of rotation); negation (multiplication by −1) leaves the magnitude unchanged but flips the axis in the opposite direction.

[5] For example, a geostationary satellite completes one orbit per day above the equator (360 degrees per 24 hours)a has angular velocity magnitude (angular speed) ω = 360°/24 h = 15°/h (or 2π rad/24 h ≈ 0.26 rad/h) and angular velocity direction (a unit vector) parallel to Earth's rotation axis (

^a Geosynchronous satellites actually orbit based on a sidereal day which is 23h 56m 04s, but 24h is assumed in this example for simplicity.

In the simplest case of circular motion at radius

from the x-axis, the orbital angular velocity is the rate of change of angle with respect to time:

is measured in radians, the arc-length from the positive x-axis around the circle to the particle is

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.

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.

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:

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.

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).

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

in this plane, as in the two-dimensional case above, one may define the orbital angular velocity vector as: where θ is the angle between r and v. In terms of the cross product, this is: From the above equation, one can recover the tangential velocity as: Given a rotating frame of three unit coordinate vectors, all the three must have the same angular speed at each instant.

In such a frame, each vector may be considered as a moving particle with constant scalar radius.

The rotating frame appears in the context of rigid bodies, and special tools have been developed for it: the spin angular velocity may be described as a vector or equivalently as a tensor.

Consistent with the general definition, the spin angular velocity of a frame is defined as the orbital angular velocity of any of the three vectors (same for all) with respect to its own center of rotation.

The addition of angular velocity vectors for frames is also defined by the usual vector addition (composition of linear movements), and can be useful to decompose the rotation as in a gimbal.

All components of the vector can be calculated as derivatives of the parameters defining the moving frames (Euler angles or rotation matrices).

The spin angular velocity vector of both frame and body about O is then where

The components of the spin angular velocity pseudovector were first calculated by Leonhard Euler using his Euler angles and the use of an intermediate frame: Euler proved that the projections of the angular velocity pseudovector on each of these three axes is the derivative of its associated angle (which is equivalent to decomposing the instantaneous rotation into three instantaneous Euler rotations).

Therefore:[7] This basis is not orthonormal and it is difficult to use, but now the velocity vector can be changed to the fixed frame or to the moving frame with just a change of bases.

are unit vectors for the frame fixed in the moving body.

[citation needed] The angular velocity tensor is a skew-symmetric matrix defined by: The scalar elements above correspond to the angular velocity vector components

The linear mapping Ω acts as a cross product