Let q be a unit quaternion which represents a rotation. Let p be a coordinate in R

^{3} which we map to H by setting the scalar to zero. Then the equation:

p' = qpq

^{-1}is p rotated by q. Multiply q by non zero scalar s.

(sq)p(sq)

^{-1} = s(qp)(s

^{-1}q

^{-1}) = (s/s)qpq

^{-1} = p'

scalars commute so the terms s and 1/s cancel so it yields the same result.

If we change the equation to using the conjugate then we get the same for unit since q

^{-1} = q

^{*} in that case.

p' = qpq

^{*}and for non unit the scale factors compose yielding the rotation plus a uniform scaling by s

^{2}:

p' = (sq)p(sq)

^{*} = s

^{2}(qpq

^{*})

If we change the mapping of p to setting the scalar to one then we are back to an unscaled rotation.

Some implications:

- All quaternions except zero can represent a rotation.
- The set of all quaternion that represent the same rotation fall on a line through the origin.
- There are only two points the line intersects the unit sphere: q and -q.
- If two quaternions don't fall on the same line through the origin then the represent different rotations.