anchor.diff:  :stem: latexmath
   :figure-caption!:
   
  -=== Before We Start
   
   Quaternion is widely used in game engines to represent 3D rotation. As a game engineer you might be using quaternion explicitly or implicitly in your daily work, but do you really understand what is going on under the hood when you are calling “rotate a vector” or “combine two rotations”? Why rotating a vector stem:[v] by quaternion stem:[q] is calculated by a “sandwich” multiplication: stem:[qvq^{-1}] ? Why rotating by quaternion stem:[q_1] then stem:[q_2] is in the reversed order: stem:[{q_2}{q_1}], and can you visualize the result rotation axis and angle?
   
  
positive.code: It becomes harder to visualize stem:[\vec{a}] and stem:[\vec{b}] if we go this way, and the negative sign does not really introduce a lot of difficulties, so we will stick with that negative sign in this article.
  
  ****
  
  
  === Flip Vector
  
negative.code: 
  *_Any 3D rotation is equivalent a unit quaternion stem:[q] that stem:[\left|q\right|=1]._*
  
  ---
  .Side Note
  ****
  This claim actually has 2 sides. Let me go a little be more in details in math term: 
  