Pure and unit quaternions
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Pure quaternion
A quaternion whose vector part is zero equals a real number corresponding to the scalar part.
A quaternion whose scalar part is zero, is called a pure quaternion:
The square of a pure quaternion is always real and not positive:
Multiplication of pure quaternions leads to the following simplified equation (for the general equation see chapter Multiplication of quaternions):
Unit quaternion
A unit quaternion, also called normalized quaternion, has a magnitude of 1:
A unit quaternion can be created from any quaternion by dividing it and so the four components by its norm:
The product of two unit quaternions and the inverse of a unit quaternion are again unit quaternions.
If is a unit quaternion, its inverse equals its conjugate: