anchor.diff:  (2). For any unit quaternion stem:[q=(x,y,z,w)], it describes a rotation of angle stem:[θ=2cos^{-1}w] along axis stem:[\vec{v}=\frac{(x,y,z)}{\sqrt{1-w^{2}}}].
   
   ****
  +---
   
   From now on, any quaternion stem:[q] used in this article is by default a unit quaternion, and we will use stem:[q] to describe rotations.
   
  
positive.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: 
  
negative.code: :stem: latexmath
  :figure-caption!:
  
  
  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?
  
  