Work out quaternion multiplication instantly with clear inputs, formula shown and shareable results.
Quaternion multiplication follows i² = j² = k² = ijk = −1 and is not commutative. The norm is multiplicative, and unit quaternions represent three-dimensional rotations without the gimbal lock that afflicts Euler angles.
Hamilton product
w = w₁w₂ − x₁x₂ − y₁y₂ − z₁z₂, with the vector parts following the cross-product pattern
−60 + 12i + 30j + 24k, with norm √30 × √174 matching the product's norm exactly.
They interpolate rotations smoothly, avoid gimbal lock and need only four numbers instead of a nine-entry matrix.