Difference between revisions of "Addition of quaternions"
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{{Navigation|before=[[Pure and unit quaternions]]|overview=[[Quaternions]]|next=[[Multiplication of quaternions]]}} | {{Navigation|before=[[Pure and unit quaternions]]|overview=[[Quaternions]]|next=[[Multiplication of quaternions]]}} | ||
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| + | Two quaternions can easily be added by adding their components. | ||
| + | |||
| + | Assume two quaternions: | ||
| + | :<math> | ||
| + | \begin{align} | ||
| + | q &= q_0 + q_1i + q_2j + q_3k \\ | ||
| + | p &= p_0 + p_1i + p_2j + p_3k | ||
| + | \end{align} | ||
| + | </math> | ||
| + | |||
| + | The quaternions <math>q</math> and <math>p</math> can just be replaced by these equations in their addition <math>q+p</math> and then be summarized as the individual four components: | ||
| + | :<math> | ||
| + | \begin{align} | ||
| + | q+p &= q_0 + q_1i + q_2j + q_3k + p_0 + p_1i + p_2j + p_3k \\ | ||
| + | &= (q_0+p_0) + (q_1+p_1)i + (q_2+p_2)j + (q_3+p_3)k | ||
| + | \end{align} | ||
| + | </math> | ||
| + | So the addition in vector notation can be written as: | ||
| + | :<math> | ||
| + | \begin{align} | ||
| + | q+p &= q_0+p_0 \oplus \vec{\mathbf{q}}+\vec{\mathbf{p}} | ||
| + | \end{align} | ||
| + | </math> | ||
| + | Obviously addition of two quaternions is ''commutative'' and ''associative'': | ||
| + | :<math> | ||
| + | \begin{align} | ||
| + | q+p&=p+q \\ | ||
| + | (q+p)+r&=q+(p+r) | ||
| + | \end{align} | ||
| + | </math> | ||
| + | |||
| + | [[Category:Article]] | ||
| + | [[Category:Quaternion]] | ||
Latest revision as of 16:24, 4 September 2015
| ← Back: Pure and unit quaternions | Overview: Quaternions | Next: Multiplication of quaternions → |
Two quaternions can easily be added by adding their components.
Assume two quaternions:
The quaternions 
and 
can just be replaced by these equations in their addition 
and then be summarized as the individual four components:
So the addition in vector notation can be written as:
Obviously addition of two quaternions is commutative and associative:
