That said when quaternions are used in geometry, it is more convenient to define them as a scalar plus a vector.
An equivalent definition of Quaternion ( R3 means 3D euclidean real vector space):
Q={q=<s,v>|s∈R,v∈R3} 1. <s,0⃗ >=s⇒R⊆Q 2. <0,v>=v⇒R3⊆Q 3. ∴<0,0⃗ >=0=0⃗Binary Operation +:Q×Q→Q
<a,u>+<b,v>=<a+b,u+v> <script type="math/tex; mode=display" id="MathJax-Element-30"> + = </script> 1. s+v=<s,0⃗ >+<0,v>=<s,v> ∴Q={q=s+v|s∈R,v∈R3} 2. <a,u>+<b,v>=<b,v>+<a,u> ∴s+v=v+s=<s,v> 3. (Q;+) is an Abelian Group.Binary Operation ∘ : Q×Q→Q
(a+u)∘(b+v)=a∘(b+v)+u∘(b+v)=a∘b+a∘v+u∘b+u∘v=ab+av+bu+u×v−u⋅v=ab−u⋅v+av+bu+u×v(×Cross Product⋅Dot Product) Absorbing element of operator ∘ : 0 p,q∈Q,p≠0∧q≠0⇒p∘q≠0 p=(a+u)q=(b+v)p∘q=0⎫⎭⎬⎪⎪⇒u×v=0⇒...⇒p=0∨q=0 ∘ is associative (a+u)∘(b+v)∘(c+w)=a∘b∘c+a∘b∘w+a∘v∘c+a∘v∘w+u∘b∘c+u∘b∘w+u∘v∘c+u∘v∘w u∘v∘w=(xi+yj+zk)∘(x′i+y′j+z′k)∘(x′′i+y′′j+z′′k) Identity element of ∘ : 1Inverse of q (Q−{0};∘) is a GroupThus, (Q;+;∘) is a Division Ring
The Next: (Rodrigues’) Rotation Formula
