How this calculator works
Quaternions extend complex arithmetic with three imaginary basis elements. Hamilton multiplication is associative but not commutative, so operand order matters. The inverse is the conjugate divided by squared magnitude. A unit rotation quaternion combines half the rotation angle with a normalised three-dimensional axis; its norm is one. This tool uses the scalar-first convention.
Formula
i^2 = j^2 = k^2 = ijk = -1. q^(-1) = conjugate(q)/|q|^2. Rotation q = cos(theta/2) + u*sin(theta/2), where u is a unit axis.
Worked example
i times j equals k, while j times i equals -k. For q = 1 + i + j + k, the squared norm is 4 and q inverse is (1 - i - j - k)/4. A half-turn about z gives the unit quaternion k.
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Frequently asked questions
Which division order is used?
p/q means p multiplied on the right by inverse(q). Multiplying the inverse on the left can give a different answer.
Must the axis already be a unit vector?
No. Rotation mode normalises a non-zero axis before applying the half-angle formula.
Does normalisation preserve an arbitrary quaternion value?
It preserves direction in four dimensions but changes magnitude to one. Use it for rotation quaternions only when that interpretation is intended.