anchor.diff:  [stem]
   ++++
   \begin{align*}
  -q_{(X,θ)}=(x,y,z,w)\cos\frac{θ}{2}+(w,-z,y,-x)\sin\frac{θ}{2})\\
  -q_{(Y,θ)}=(x,y,z,w)\cos\frac{θ}{2}+(z,w,-x,-y)\sin\frac{θ}{2})\\
  -q_{(Z,θ)}=(x,y,z,w)\cos\frac{θ}{2}+(-y,x,w,-z)\sin\frac{θ}{2})\\
  -\end{align*}
  -++++  
  -
  -=== Appendix: Derive Quaternion Multiplication
  +q_{(X,θ)}=(x,y,z,w)\cos\frac{θ}{2}+(w,-z,y,-x)\sin\frac{θ}{2}\\
  +q_{(Y,θ)}=(x,y,z,w)\cos\frac{θ}{2}+(z,w,-x,-y)\sin\frac{θ}{2}\\
  +q_{(Z,θ)}=(x,y,z,w)\cos\frac{θ}{2}+(-y,x,w,-z)\sin\frac{θ}{2}\\
  +\end{align*}
  +++++  
  +
  +== 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.
   
  
positive.code: [stem]
  ++++
  \begin{align*}
  q_{(X,θ)}=(x,y,z,w)\cos\frac{θ}{2}+(w,z,-y,-x)\sin\frac{θ}{2})\\
  q_{(Y,θ)}=(x,y,z,w)\cos\frac{θ}{2}+(-z,w,x,-y)\sin\frac{θ}{2})\\
  q_{(Z,θ)}=(x,y,z,w)\cos\frac{θ}{2}+(y,-x,w,-z)\sin\frac{θ}{2})\\
  \end{align*}
  ++++  
  
  
negative.code: q_Z=q'q=((0,0,1),0)((x,y,z),w)=((-y,x,w),-z)
  ++++
  
  We can use the same method to generalize the angle to stem:[θ], and let stem:[q'=((0,0,\sin\frac{θ}{2}),\cos\frac{θ}{2})],
  
  [stem]
  ++++
  