Remake Quaternions

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=0⃗ 

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Binary Operation +:Q×Q→Q

<a,u>+<b,v>=<a+b,u+v>

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.

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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)

1. Absorbing element of operator ∘ : 0
2. 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
3. ∘ 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)
4. Identity element of ∘ : 1
5. Inverse of q
6. (Q−{0};∘) is a Group
7. Thus, (Q;+;∘) is a Division Ring

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The Next: (Rodrigues’) Rotation Formula