What does this quaternion calculator do?

This calculator performs core quaternion operations and a practical 3D vector rotation mode. A quaternion is written as a+bi+cj+dk, where a is the real part and b, c, d are the coefficients of the i, j, and k directions.

• q1+q2 and q1−q2: component-wise addition and subtraction.
• q1×q2: Hamilton product, where order matters.
• Conjugate, norm and inverse: operations applied to q1.
• q1÷q2: interpreted as q1×q2⁻¹, so q2 must not be the zero quaternion.
• Rotate mode: rotates a 3D vector by q1 as a unit quaternion, with auto-normalization warning when needed.

What is a quaternion?

A quaternion is an extension of complex numbers with one real component and three imaginary-direction components. Complex numbers use a+bi; quaternions use a+bi+cj+dk. The i, j, and k units follow special multiplication rules, which make quaternion multiplication powerful but order-sensitive.

For example, i×j=k, but j×i=−k. This is the simplest way to see why Hamilton product is not ordinary commutative multiplication.

• Real part: a.
• i component: b.
• j component: c.
• k component: d.
• Vector part: bi+cj+dk.

Hamilton product explained

The Hamilton product is the standard multiplication rule for quaternions. The calculator combines the four components of q1 and q2 using the multiplication identities for i, j and k, then returns another quaternion in a+bi+cj+dk form.

1. The real and vector parts are combined together.
2. The rules i²=j²=k²=−1 are applied.
3. The direction rules i×j=k, j×k=i and k×i=j are preserved.
4. Reverse-order products may change signs, so the final components can be different.

Conjugate, norm, inverse and division

The conjugate changes the signs of the i, j and k components. The norm measures the size of the quaternion. The inverse uses the conjugate and norm squared; it is defined only when the quaternion is not zero.

• Conjugate: turns a+bi+cj+dk into a−bi−cj−dk.
• Norm: a real magnitude value, shown approximately.
• Inverse: q⁻¹, blocked for the zero quaternion.
• Division: q1÷q2 is computed as q1×q2⁻¹, not component-by-component division.

3D vector rotation with a quaternion

In rotate mode, the vector v=(x,y,z) is treated as a pure quaternion, and the result is computed using the sandwich product q v q⁻¹. This is the standard quaternion rotation pattern when q is a unit quaternion.

If q1 is not unit length, the calculator normalizes it first and displays a warning. This prevents the rotation from being distorted by scale while keeping the user aware that the entered q1 was not already a unit quaternion.

Worked examples

Example 1: if q1=i and q2=j, the Hamilton product is k. If the order is reversed, j×i=−k. This example captures the non-commutative nature of quaternion multiplication.

Example 2: for q=1+2i+3j+4k, the conjugate is 1−2i−3j−4k. Norm and inverse are displayed as approximate numeric values where needed.

Example 3: a unit quaternion representing a 90° rotation around the z-axis sends the vector (1,0,0) approximately to (0,1,0). The calculator reports the rotated vector triple, not a rendered 3D scene.

Common mistakes

• Treating Hamilton product as commutative multiplication.
• Using a non-unit quaternion for rotation without noticing normalization.
• Thinking quaternion division means component-wise division.
• Trying to invert the zero quaternion.
• Expecting a matrix, Euler angles, or an interactive 3D plot from this calculator.

Limitations

This calculator is intentionally focused on quaternion arithmetic and vector rotation. It does not include matrix representation, Euler-angle conversion, SLERP interpolation, symbolic quaternion manipulation, arbitrary-precision arithmetic, or an interactive 3D visualization.

• No matrix output.
• No Euler-angle conversion.
• No SLERP/interpolation mode.
• No symbolic simplification.
• Approximate operations are limited by normal floating-point precision.