In the last equation, the cross product is a vector, and the dot product is a scalar. How do we subtract a scalar from a vector?
@Ken Matrix representation?
How we get the uxv-u.v expression tho?
This is a simplification of the quaternion product from the equation above, where both a=0 and b=0. So, plugging this into the equation above, we get the following expression:
(0,u)(0,v) = (-dot(u,v), cross(u,v))
In other words, the scalar resulting from the dot product is the real-valued component, and the cross product represents the three imaginary components of our quaternion. The expression at the bottom of the slide is representing this in short form (I agree it is a bit confusing..).