Multiply quaternions, with norms, dot product, conjugate and inverse of the first quaternion.
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Result
q1 × q2
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Norm |q1|
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Norm |q2|
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Dot product
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Conjugate of q1
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How to use this tool
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Enter q1 componentsType w, x, y, z for the first quaternion.
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Enter q2 componentsType w, x, y, z for the second.
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Read the productNote the order: q1 × q2.
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Read norms and conjugateUse the conjugate to form the inverse.
Formula & worked example
Product: (w1w2 − x1x2 − y1y2 − z1z2, w1x2 + x1w2 + y1z2 − z1y2, w1y2 − x1z2 + y1w2 + z1x2, w1z2 + x1y2 − y1x2 + z1w2).
i × j = k: (0, 1, 0, 0) × (0, 0, 1, 0) = (0, 0, 0, 1).
Limitations — read this before you rely on it
Pure quaternion arithmetic; it does not convert to rotation matrices or Euler angles.
Frequently asked questions
A four-part number w + xi + yj + zk, extending complex numbers to 3-D rotations.
No: q1 × q2 generally differs from q2 × q1.
Imaginary units with i² = j² = k² = ijk = −1, and ij = k, jk = i, ki = j.
√(w² + x² + y² + z²), like a 4-D length.
The same real part with negated imaginary parts; q × conjugate(q) = |q|².