There are four terms: an identity term and three rotations. Q = a + bi + cj + dk = a I + b R_yz + c R_zx + d R_xy = aI + v·R, where v is the axis of rotation.

There are four terms: an identity term and three rotations. Q = a + bi + cj + dk = a I + b R_yz + c R_zx + d R_xy = aI + v·R, where v is the axis of rotation. Basically you have an extra degree of freedom because you can multiply the whole thing by a scalar that just scales whatever vector it multiplies. But if you just want to represent rotations, you want it to have unit magnitude, so you ensure that a = cos θ and |v| = sin θ. (or θ/2 on each if you want to use the two-sided qpq^{-1} rotation).