# Quaternion to euler online dating

I have a vector in 3D space (red vector $r$) where the origin of the vector is placed at the origin of the coordinate system and the other edge of the vector is placed at the surface of a unit sphere.

I rotate the vector around the origin of the coordinate system and using motion sensors, I capture the rotation in the form of a quaternion.

However, here we're attempting to generalize complex numbers, not cross products per se.

So, instead of assigning different normal vectors to each cross product term, let's assign a different complex number to each term. Then, we assign to the cross product of two vectors in the -plane and to the cross product of two vectors in the -plane.

Quaternions seem to be one of the least understood mathematical things amongst physicists.the $i,j,k$ are treated like the hypercomplex parts of the quaternion rather than just geometrically).Then you have $$ r' = qrq^* $$ using quaternion multiplication.However, they usually give less-than-satisfactory attempts at generalizing, highlighting the mysterious algebraic problem of ``closure'' or something to that effect.Then, often retelling the story of Hamilton and a bridge they pull some strange, ``4D'' quaternions out of a hat and show how they happily resolve all the algebraic problems.