Difference between revisions of "Pure and unit quaternions"
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A quaternion whose scalar part is zero, is called a pure quaternion: | A quaternion whose scalar part is zero, is called a pure quaternion: | ||
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:<math> | :<math> | ||
− | + | Re(q) = 0 | |
</math> | </math> | ||
− | The square of pure | + | |
+ | The square of a pure quaternion is always real and not positive: | ||
:<math> | :<math> | ||
q^2 \in \R, q^2\le 0 | q^2 \in \R, q^2\le 0 | ||
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q^{-1}=q^* | q^{-1}=q^* | ||
</math> | </math> | ||
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+ | [[Category:Article]] | ||
+ | [[Category:Quaternion]] |
Latest revision as of 15:59, 8 September 2015
← Back: Basic properties of quaternions | Overview: Quaternions | Next: Addition of quaternions → |
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: