Difference between revisions of "Pure and unit quaternions"
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A quaternion whose vector part is zero equals a real number corresponding to the scalar part. | 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 | + | A quaternion whose scalar part is zero, is called a pure quaternion: |
| + | :<math> | ||
| + | Re(q) = 0 | ||
| + | </math> | ||
| + | |||
| + | The square of a pure quaternion is always real and not positive: | ||
| + | :<math> | ||
| + | q^2 \in \R, q^2\le 0 | ||
| + | </math> | ||
| + | |||
| + | Multiplication of pure quaternions leads to the following simplified equation (for the general equation see chapter [[Multiplication of quaternions]]): | ||
| + | :<math> | ||
| + | (0,\vec{\mathbf{x}})(0,\vec{\mathbf{y}})=(-\vec{\mathbf{x}}\cdot\vec{\mathbf{y}},\vec{\mathbf{x}}\times\vec{\mathbf{y}}) | ||
| + | </math> | ||
| − | + | ===Unit quaternion=== | |
| + | A unit quaternion, also called normalized quaternion, has a magnitude of 1: | ||
| + | :<math> | ||
| + | |q| = q_0^2 + q_1^2 + q_2^2 + q_3^2 = 1 | ||
| + | </math> | ||
| + | A unit quaternion can be created from any quaternion by dividing it and so the four components by its norm: | ||
:<math> | :<math> | ||
| − | + | q_u = \frac{q}{|q|} = \frac{q_0}{|q|} + \frac{q_1}{|q|}i + \frac{q_2}{|q|}j + \frac{q_3}{|q|}k | |
</math> | </math> | ||
| − | + | The product of two unit quaternions and the inverse of a unit quaternion are again unit quaternions. | |
| + | |||
| + | If <math>q</math> is a unit quaternion, its [[Basic properties of quaternions|inverse]] equals its [[Basic properties of quaternions|conjugate]]: | ||
:<math> | :<math> | ||
| − | + | q^{-1}=q^* | |
| − | |||
| − | |||
| − | |||
| − | |||
</math> | </math> | ||
| + | |||
| + | [[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:
