Versor

Versors are an algebraic parametrisation of rotations. In classical quaternion theory a versor is a quaternion of norm one (a unit quaternion).

Each versor has the form

where the r² = −1 condition means that r is a unit vector (in 3-dimensions). In case a = π/2, the versor is termed a right versor.

The corresponding 3-dimensional rotation has the angle 2a about the axis r in axis–angle representation.

The word is derived from Latin versare = "to turn" with the suffix -or forming a noun from the verb (i.e. versor = "the turner"). It was introduced by William Rowan Hamilton in the context of his quaternion theory. For historical reasons, it sometimes is used synonymously with a "unit quaternion" without a reference to rotations.

In the quaternion algebra a versor ${\displaystyle q=\exp(a\mathbf {r} )}$ will rotate any quaternion v through the sandwiching product map ${\displaystyle v\mapsto qvq^{-1}}$ such that the scalar part of v is preserved. If this scalar part (the fourth dimension of the quaternion space) is zero, i.e. v is a Euclidean vector in three dimensions, then the formula above defines the rotation through the angle 2a around the unit vector r. For this case, this formula expresses the adjoint representation of the Spin(3) Lie group in its respective Lie algebra of 3-dimensional Euclidean vectors, and the factor "2" is due to the double covering of Spin(3) over the rotation group SO(3). In other words, qvq⁻¹ rotates the vector part of v around the vector r. See quaternions and spatial rotation for details.

...
Wikipedia