anchor.diff:  [stem]
   ++++
   \begin{align*}
  -q&=(sin⁡\frac{θ}{2}\vec{v},cos⁡\frac{θ}{2})\\
  +q&=(\sin⁡\frac{θ}{2}\vec{v},\cos⁡\frac{θ}{2})\\
   &=-{q_a}{q_v}\\
   &=-{q_v'}{q_a}
   \end{align*}
  
positive.code: 
  *_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]._*
  
  
negative.code: {q_1}{q_2}=(\vec{v_1},w_1)(\vec{v_2},w_2)=(w_1\vec{v_2} + w_2\vec{v_1} + \vec{v_1}×\vec{v_2}, {w_1}{w_2}-\vec{v_1}·\vec{v_2})
  ++++
  
  Note here stem:[q_1] and stem:[q_2] are not necessarily unit quaternion, so even I’m using vector form, there’s no need to put stem:[\sin⁡\frac{θ}{2}] and stem:[\cos⁡\frac{θ}{2}] as we did for unit quaternions. It’s hard to explain this definition without introducing the algebra structure of quaternions, so I will skip that. If you are interesting to know how this is derived, quaternion https://en.wikipedia.org/wiki/Quaternion#Definition[Wiki page] has a very straight forward introduction.
  
  We are not going to use this general quaternion multiplication in Part 1. Here we only need to know a simpler form, the multiplication of flips. Let stem:[q_a=(\vec{a},0)], stem:[q_b=(\vec{b},0)] then
  
  