Quaternion Multiplication Calculator

Work out quaternion multiplication instantly with clear inputs, formula shown and shareable results.

About Quaternion Multiplication Calculator

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.

Formula

Hamilton product

w = w₁w₂ − x₁x₂ − y₁y₂ − z₁z₂, with the vector parts following the cross-product pattern

Worked example

With the default values below, Quaternion Multiplication Calculator returns Product q₁q₂ of -60 + 12i + 30j + 24k. Change any input above and every figure here is recalculated in your browser.

Inputs used

First quaternion w, x, y, z: 1, 2, 3, 4
Second quaternion w, x, y, z: 5, 6, 7, 8

Results

Product q₁q₂: -60 + 12i + 30j + 24k
Norm of the product: 72.24956747
|q₁| × |q₂| (must match): 72.24956747
Scalar part of the product: -60.000000

Step by step

1Values used
2Hamilton product
3Product q₁q₂= -60 + 12i + 30j + 24k
4Norm of the product= 72.24956747
5|q₁| × |q₂| (must match)= 72.24956747
6Scalar part of the product= -60.000000

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Multiply 1+2i+3j+4k by 5+6i+7j+8k.

−60 + 12i + 30j + 24k, with norm √30 × √174 matching the product's norm exactly.

Why do games use quaternions?

They interpolate rotations smoothly, avoid gimbal lock and need only four numbers instead of a nine-entry matrix.