Quaternion calculator

Quaternions extend complex numbers to four parts: a scalar w and a vector x, y, z carrying the i, j, k units. Addition runs componentwise, but multiplication does not: the Hamilton product folds each part of one quaternion into each part of the other, so a·b and b·a usually differ, and both are shown with their order labeled. The conjugate flips the vector part, the norm measures length through space of four dimensions, and the inverse is the conjugate divided by the norm squared, which is why only the zero quaternion has no inverse. These are the raw algebra results; the tool does not convert to rotation matrices or Euler angles, so wiring this into a 3D rotation pipeline is on you.