What does it mean by uv = u x v - u · v in R3 case? I thought u x v is a 3-by-1 vector and u · v is a scalar. How could them be combined together?

@yongchi1 $u \times v$ can be interpreted as a linear combination of $i, j, k$, and $-u · v$ makes up the real part.

So, to clarify, the quaternion product uv would then have scalar part -dotp (u,v) and vector part crossp (u,v)?

What does it mean by uv = u x v - u · v in R3 case? I thought u x v is a 3-by-1 vector and u · v is a scalar. How could them be combined together?

@yongchi1 $u \times v$ can be interpreted as a linear combination of $i, j, k$, and $-u · v$ makes up the real part.

So, to clarify, the quaternion product uv would then have scalar part -dotp (u,v) and vector part crossp (u,v)?