anchor.diff:  ****
   
   
  -=== Flip Vector
  +== Flip Vector
   
   Given a flip stem:[q_a=(\vec{a},0)] and vector stem:[\vec{v}], we are ready to calculate the result of the flip stem:[\vec{v'}].
   
  
positive.code: 
  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.
  
  === Quaternion and Flip
  
  Now let’s recall the quaternion vector form stem:[q=(sin⁡\frac{θ}{2}\vec{v},cos⁡\frac{θ}{2})]. With the discussion of flips above, you can almost immediately see why we are using stem:[\frac{θ}{2}] here.
  
  
negative.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]
  