anchor.diff:  
   Now we can rewrite our flip composition rule in a more specific form:
   
  -*_Any 3D rotation equivalent to rotating angle stem:[\textbf{θ}] along axis stem:[\textbf{\vec{v}}] can be represented as a sequence of 2 flips along axis stem:[\textbf{\vec{a}}] and stem:[\textbf{\vec{b}}], such that stem:[\textbf{\vec{a}·\vec{v}=0}], stem:[\textbf{\vec{b}·\vec{v}=0}] and the angle from stem:[\textbf{\vec{a}}] to stem:[\textbf{\vec{b}}]: stem:[\textbf{<\vec{a},\vec{b}>=\frac{θ}{2}}]._*
  +*_Any 3D rotation equivalent to rotating angle stem:[θ] along axis stem:[\vec{v}] can be represented as a sequence of 2 flips along axis stem:[\vec{a}] and stem:[\vec{b}], such that stem:[\vec{a}·\vec{v}=0], stem:[\vec{b}·\vec{v}=0] and the angle from stem:[\vec{a}] to stem:[\vec{b}]: stem:[<\vec{a},\vec{b}>=\frac{θ}{2}]._*
   
   This representation means if we fully understand flip, which is easier to visualize, we can fully understand rotation and quaternions, since any quaternion can be broken down to flips.
   
  
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:[\textbf{θ}] along some axis stem:[\textbf{\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:[\textbf{q}] that stem:[\textbf{\left|q\right|=1}]._*
  
  ---
  [NOTE, indent=2]
  
negative.code: 
  With flip multiplication defined, we can rewrite our flip composition rule again:
  
  *_Any 3D rotation stem:[q=(sin⁡\frac{θ}{2}\vec{v},cos⁡\frac{θ}{2})] can be represented as a sequence of 2 flips stem:[q_a=(\vec{a},0)] and stem:[q_b=(\vec{b},0)], such that_*
  [stem]
  ++++
  q=-{q_b}{q_a}
  ++++
  *_where stem:[\vec{a}·\vec{v}=0], stem:[\vec{b}·\vec{v}=0] and the angle from stem:[\vec{a}] to stem:[\vec{b}]: stem:[<\vec{a},\vec{b}>=\frac{θ}{2}]._*
  
  You might be thinking why it is not stem:[q= {q_a}{q_b}] instead. We will show where the order and the negative sign coming from in the proof.
  
  