Quaternion Calculator

Quaternion arithmetic for 3D rotation

(1 + 2i + 3j + 4k) ⊗ (5 + 6i + 7j + 8k) | i² = j² = k² = ijk = −1

Result R

−60 + 12i + 30j + 24k

Norm |R|

72.2496

Not a unit quaternion — normalized below to read the rotation

Equivalent 3D rotation

Rotation angle θ

292.291° (5.10144 rad)

Rotation axis (x, y, z)

(0.2981, 0.7454, 0.5963)

Reverse product B ⊗ A

−60 + 20i + 14j + 32k

A ⊗ B ≠ B ⊗ A — quaternion multiplication does not commute, so rotation order matters

About this calculator

The Quaternion Calculator multiplies, adds, subtracts, and divides quaternions and reads the result back as a 3D rotation. It computes the Hamilton product A ⊗ B (and the reverse B ⊗ A to show non-commutativity), the norm |R|, the inverse-based quotient A ⊗ B⁻¹, and the equivalent axis–angle rotation θ about a unit axis. Use it for 3D graphics, robotics, aerospace attitude, and IMU/orientation math where quaternions avoid gimbal lock.

Common examples

(1 + 2i + 3j + 4k) ⊗ (5 + 6i + 7j + 8k) = −60 + 12i + 30j + 24k
Reverse order (5 + 6i + 7j + 8k) ⊗ (1 + 2i + 3j + 4k) = −60 + 20i + 14j + 32k — a different result
0.7071 + 0.7071i is a unit quaternion: a 90° rotation about the x-axis
(0 + 1i + 0j + 0k) is a 180° rotation about the x-axis
Dividing a quaternion by itself, A ⊗ A⁻¹, gives the identity 1 + 0i + 0j + 0k