# Pure and unit quaternions

From Robotics

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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: