Quaternion calculator

Quaternion 1 (w + xi + yj + zk)

Quaternion 2 (w + xi + yj + zk)

Results

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Quaternion multiplication is not commutative — q1×q2 usually differs from q2×q1 — so both orders are shown when they differ. Norm is the length sqrt(w²+x²+y²+z²); a rotation quaternion should have norm 1. The conjugate flips the sign of x, y and z; for a unit quaternion, the inverse equals the conjugate.

What a quaternion is here

Each quaternion is entered as four numbers, w, x, y and z, representing w + xi + yj + zk. This is the form used in 3D graphics, robotics and flight dynamics to represent a rotation without the axis-order ambiguity and gimbal lock that Euler angles run into.

Why multiplication needs its own calculator

Quaternion multiplication is not commutative - q1 × q2 is generally not the same as q2 × q1 - because i, j and k themselves multiply in a fixed, non-commuting pattern (ij = k, but ji = -k). Working this out by hand is where mistakes creep in; this applies the Hamilton product term by term and shows both orders when they differ.

Norm, conjugate and inverse

The norm is the quaternion's length, sqrt(w²+x²+y²+z²) - a rotation quaternion should have norm 1, and a norm far from 1 usually means accumulated floating point drift in whatever produced it. The conjugate flips the sign of x, y and z. The inverse is the conjugate divided by the squared norm, and for a unit quaternion the inverse equals the conjugate exactly - shown here so that identity can be checked rather than taken on faith.

What this is not

This is arithmetic on the four numbers, not a 3D viewer - there is no rotating cube. It is aimed at checking a hand calculation or a value copied out of code, not at visualising the rotation itself.