anchor.diff:  
   We slightly touched quaternion multiplication, and we proved an important equation stem:[{q_1^{-1}}{q_2^{-1}}=({q_2}{q_1})^{-1}].
   
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  -[NOTE, indent=2]
  -.Side Note
  -====
  +=== Appendix: Derive Quaternion Multiplication
  +
   This part is not very important for understanding quaternion. It is a bit calculation heavy and is more for fun. Feel free to skip.
   
   We can actually derive the general quaternion multiplication from this special flip break down. That is if we define flip multiplication stem:[{q_a}{q_b}=(\vec{a},0)(\vec{b},0)=(\vec{a}×\vec{b},-\vec{a}·\vec{b})] directly, we can proof what general quaternion multiplication stem:[{q_1}{q_2}=(sin⁡\frac{θ_1}{2}\vec{v_1},cos⁡\frac{θ_1}{2})(sin⁡\frac{θ_2}{2}\vec{v_2},cos⁡\frac{θ_2}{2})] would look like.
  
positive.code: 
  In order to represent 3D rotation, we have a constraint on the quaternions we use. But before that I want to introduce Euler’s rotation theorem:
  
  *_Any rotation in 3D space is equivalent to a single rotation of angle stem:[θ] along some axis stem:[\vec{v}]._*
  
  We can use quaternion to describe this angle-axis rotation : stem:[q=(sin⁡\frac{θ}{2}\vec{v}.x,sin⁡\frac{θ}{2}\vec{v}.y,sin⁡\frac{θ}{2}\vec{v}.z,cos⁡\frac{θ}{2})], or in a more compact form stem:[q=(sin⁡\frac{θ}{2}\vec{v},cos⁡\frac{θ}{2})]. We call this form the vector form of a quaternion, and we will use this form throughout this article. You might be thinking why we are using stem:[\frac{θ}{2}] other than using stem:[θ] directly. I will explain that in a later section.
  
  It is easy to see the length of this quaternion stem:[\left|q\right|=\sqrt{sin^{2}\frac{θ}{2}\left|\vec{v}\right|^{2}+cos^{2}\frac{θ}{2}}=1]. (Remember the axis stem:[\vec{v}] is a unit vector that stem:[\left|\vec{v}\right|=1]). We call it a unit quaternion if the length stem:[\left|q\right|=1]. So we can rewrite Euler’s rotation theorem in quaternion term:
  
  *_Any 3D rotation is equivalent a unit quaternion stem:[q] that stem:[\left|q\right|=1]._*
  
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  [NOTE, indent=2]
  
negative.code: ++++ 
  which is the definition of quaternion multiplication of stem:[{q_1}{q_2}=(sin⁡\frac{θ_1}{2}\vec{v_1},cos⁡\frac{θ_1}{2})(sin⁡\frac{θ_2}{2}\vec{v_2},cos⁡\frac{θ_2}{2})].
  
  