Quaternions were originally invented by Sir William Rowan Hamilton in 1843 as a generalization of complex numbers. He determined that just as in two dimensions, where complex numbers could be defined as a sum of a real and an imaginary part, a generalization could be made to four dimensions, giving quaternions.
The inner product of two quaterions.
A quaternion can be normalized by dividing by its length.
Spherical linear interpolation gives constant speed motion along great circle arcs on the surface of a sphere and consequently has two desirable properties for interpolating rotations: