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.

How to use the Quaternion Calculator calculator

Enter the values for your problem into the input fields.
Read the result — it updates instantly as you type.
Check the formula and the visual explanation to follow how the answer was found.
Copy the page URL to share the exact calculation.

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

Frequently asked questions

How does the Quaternion Calculator calculator work?

Enter your values and the calculator applies the exact mathematical method for this problem, showing the result together with the formula it used. Everything is computed with high-precision arithmetic, so the answer you see is not limited by ordinary floating-point rounding.

When would I use the Quaternion Calculator calculator?

It is useful for homework and exam preparation, for checking work done by hand, and whenever a step in a larger problem needs this calculation done quickly and reliably.

How accurate are the results?

Calculations run at 30 significant digits of precision internally. Displayed values are rounded for readability, but the underlying result is far more precise than a typical hand or pocket-calculator computation.

Can I share a specific calculation with someone else?

Yes. Every value you enter updates the URL, so copying the address bar and sharing the link reopens the exact same calculation for whoever clicks it.