anchor.diff:  \end{align*}
   ++++  
   
  -== Summary of Part 1
  +=== Summary of Part 1
   
   In Part 1, we covered the definition of quaternion stem:[q=(x,y,z,w)], the vector form of quaternion stem:[q=(\vec{v},w)], unit quaternion stem:[q=(sin⁡\frac{θ}{2}\vec{v},cos⁡\frac{θ}{2})] and how it is used to represent a rotation.
   
  
positive.code: 
  When we put a vector stem:[\vec{v}] in quaternion multiplication, we are implicitly making that vector the axis of a flip to stuff it into a quaternion stem:[(\vec{v},0)]. This is how the “sandwich” multiplication form comes from, but only in the form of flip. We will prove that our result holds the same for any rotation in the next section.
  
  == Rotate Vector
  
  We know any 3D rotation stem:[q] can be broken down into 2 flips stem:[q= -{q_b}{q_a}], which means flipping along stem:[\vec{a}] first and then stem:[\vec{b}]. So for a vector stem:[\vec{v}], we apply the first flip and get
  [stem]
  
negative.code: 
  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.
  
  === Rotation and Flip
  
  Now let’s forget quaternion for a minute, and focus on the nature of rotations. This part is the key to understand quaternion calculation in an easier way.
  
  