I don't speak well English, so please edit this question to be more accurate.
Quaternions are considered as the quotient of the 3D vectors division.
$${\bf v}\,{\bf r}^{-1}=-\frac{{\bf v}\,{\bf r}}{r^2}=-\frac{1}{r^2}(-{\bf v}\cdot{\bf r}+{\bf v}\times{\bf r})$$
but in its definition we have dot product and cross product that depends on the angle between the vectors.
To make quaternions more rational, I would like to apply rational trigonometry's concepts, quadrance and spread, to avoid using transcendental functions sine and cosine.
So, can we implement a quaternion/division-algebra by applying quadrance and spread instead of magnitude and angle?