Enter two quaternions q1 and q2 (each with components w, x, y, z) and choose an operation — addition, subtraction, multiplication, division, dot product, magnitude, conjugate, inverse, or normalization. The Quaternion Calculator returns the result quaternion along with its magnitude and a visual breakdown of components.
Result Quaternion
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Result w
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Result x
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Result y
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Result z
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Magnitude |result|
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A quaternion is an extension of complex numbers with one real (scalar) component w and three imaginary components x, y, and z. It takes the form q = w + xi + yj + zk, where i, j, k are distinct imaginary units satisfying i² = j² = k² = ijk = −1. Quaternions were introduced by Sir William Rowan Hamilton in 1843.
To rotate a 3D vector v by angle θ around a unit axis (ux, uy, uz), construct the unit quaternion q = cos(θ/2) + sin(θ/2)(ux·i + uy·j + uz·k). The rotated vector is obtained by computing q × p × q⁻¹, where p = (0, vx, vy, vz) is the vector expressed as a pure quaternion. Unit quaternions avoid gimbal lock and interpolate smoothly using SLERP.
Quaternion multiplication follows the Hamilton product. For q1 = (w1, x1, y1, z1) and q2 = (w2, x2, y2, z2), the product w = w1w2 − x1x2 − y1y2 − z1z2; x = w1x2 + x1w2 + y1z2 − z1y2; y = w1y2 − x1z2 + y1w2 + z1x2; z = w1z2 + x1y2 − y1x2 + z1w2. Note that quaternion multiplication is NOT commutative: q1 × q2 ≠ q2 × q1 in general.
Yes. Every 3D rotation can be represented by a unit quaternion (magnitude = 1), and every unit quaternion represents a valid 3D rotation. However, there is a double-cover property: both q and −q represent the same rotation, so the mapping is 2-to-1.
No. Quaternion multiplication is non-commutative, meaning q1 × q2 ≠ q2 × q1 in general. This distinguishes them from real and complex numbers. However, quaternion addition is commutative: q1 + q2 = q2 + q1.
The inverse of a quaternion q = (w, x, y, z) is q⁻¹ = conjugate(q) / |q|². The conjugate is (w, −x, −y, −z), and |q|² = w² + x² + y² + z². For a unit quaternion, the inverse equals the conjugate since |q|² = 1.
The magnitude of quaternion q = (w, x, y, z) is |q| = √(w² + x² + y² + z²). A quaternion with magnitude 1 is called a unit quaternion or versor. Normalizing a quaternion divides each component by its magnitude to produce a unit quaternion.
Quaternions form a division ring (also called a skew field), not a field. This is because multiplication is non-commutative. However, every non-zero quaternion has a multiplicative inverse, and all other field axioms (associativity, distributivity, identity elements) hold.