Quaternion Calculator

A quaternion \(q\) is a number of the form \(w + xi + yj + zk\). It consists of a scalar part (\(w\)) and a vector part (\(xi + yj + zk\)).

Conjugate (\(q^*\)): Negate the vector part. \(q^* = w - xi - yj - zk\)

Magnitude (\(|q|\)): \(\sqrt{w^2 + x^2 + y^2 + z^2}\)

Inverse (\(q^{-1}\)): \(\frac{q^*}{|q|^2}\)

3D Rotation: To rotate a vector \(v\) by an angle \(\theta\) around an axis \(u\), you first create a rotation quaternion \(q\). The vector \(v\) is represented as a pure quaternion \(p=(0, v_x, v_y, v_z)\). The rotated vector \(v'\) is the vector part of the result \(p' = qpq^*\).