Quaternion Multiplication Calculator

About this calculator

Quaternions extend complex numbers with three imaginary units (i, j, k) instead of one, and are written as q = w + xi + yj + zk. Multiplying two quaternions uses the Hamilton product, a non-commutative operation (qp ≠ pq in general) defined by the rules i² = j² = k² = ijk = −1, which combines the scalar and vector parts of both quaternions into a new quaternion.

This calculator takes the four components of two quaternions, computes their Hamilton product, and reports the resulting quaternion along with its magnitude and scalar part. Quaternions are the standard way to represent 3D rotations in computer graphics, robotics, flight simulation, and game engines because, unlike Euler angles, they avoid gimbal lock and compose smoothly through multiplication.