What is "versor"

Versor \Ver"sor\, n. [NL., fr. L. vertere, versus, to turn. See Version.] (Geom.) The turning factor of a quaternion.

Note: The change of one vector into another is considered in quaternions as made up of two operations; 1st, the rotation of the first vector so that it shall be parallel to the second; 2d, the change of length so that the first vector shall be equal to the second. That which expresses in amount and kind the first operation is a versor, and is denoted geometrically by a line at right angles to the plane in which the rotation takes place, the length of this line being proportioned to the amount of rotation. That which expresses the second operation is a tensor. The product of the versor and tensor expresses the total operation, and is called a quaternion. See Quaternion.

Quadrantal versor. See under Quadrantal.

n. 1 (context quaternion algebra English) A quaternion of norm one 2 (context geometric algebra English) A multivector representing a proper or improper rotation

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

q = exp(ar) = cosa + rsina,  r =  − 1,  a ∈ [0, π], 
where the r = −1 condition means that r is a 3-dimensional unit vector. In case , the versor is termed a right versor.

The corresponding 3-dimensional rotation has the angle a 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.

Versor may refer to:

• Versor, a quaternion of norm 1
• Versor (physics), a unit vector codirectional with an axis or with another vector
• A rotor (mathematics) or an improper rotation, a multivector in geometric algebra
• A hyperbolic versor, a generalization of quaternionic versors

In geometry and physics, the versor of an axis or of a vector is a unit vector indicating its direction.

The versor of a Cartesian axis is also known as a standard basis vector. The versor of a vector is also known as a normalized vector.