Tốc độ chuyển động thích hợp cao có thể chỉ ra rằng một ngôi sao nằm gần đó, vì các ngôi sao xa hơn phải di chuyển với vận tốc cao hơn để đạt được cùng tốc độ di chuyển góc trên thiên cầu.
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, 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 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 ). Taking polar coordinates 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:
Note that the above expression for is only valid if is in the same plane as the motion