I'm a little confused as to what q is. Is q just the the representation of the rotation around u by angle theta? What would q like in this case?

I also found this confusing: How do we actually interpret the expression of "q_bar times q"? Is it like the arch length?

What is the difference geometrically between qbar and q? And mathematically, how to compute qbar times q, we just treat qbar as q?

Is q bar the conjugate complex of q? What is q bar for sure?