Quaternion Calculator

How to use this quaternion calculator

What this calculator does

This calculator evaluates common quaternion algebra and rotation operations. A quaternion is written as q = a + bi + cj + dk, with one real coefficient and three imaginary-basis coefficients. The tool can add or subtract two quaternions, calculate both multiplication orders, divide using the inverse of q₂, find the magnitude, conjugate, inverse, or 4 × 4 real matrix of q₁, construct a unit quaternion from an axis and angle, and rotate a three-dimensional vector. It reports exact numerical identities for the entered values, but it does not determine whether a chosen coordinate-frame convention is appropriate for a particular engineering or graphics system.

When to use it

Use the calculator to check hand calculations in algebra courses, compare q₁ × q₂ with q₂ × q₁, verify an inverse or norm identity, convert an axis-angle description into a rotation quaternion and 3 × 3 matrix, or test how a vector changes under a specified 3D rotation. For deeper background, the Wolfram MathWorld quaternion overview summarizes the algebra and Hamilton's multiplication rules.

How to calculate

1. The calculator opens with a complete product example: q₁ = 2 – i + 3j + k and q₂ = 5 – 4i + k. Its result and a validated example workbook are available immediately.
2. Choose an operation in I want to find the.... The relevant input groups appear automatically.
3. Replace the displayed coefficients or rotation values. Use a decimal point, optional leading sign, and ordinary decimal notation; commas and scientific notation are rejected to prevent ambiguous parsing.
4. Read Primary result, the optional Secondary result, the four norm/scalar cards, and the adaptive Component details table. Matrix-producing operations also show Matrix representation.
5. Select Download Excel to export the current validated inputs and canonical outputs. Reset clears the demonstration and all semantic data, so Excel export becomes unavailable until a complete valid state is entered again.

Input guide

I want to find the... is required and selects the formula. For binary algebra, Real part (a), The i coefficient (b), The j coefficient (c), and The k coefficient (d) define q₁; Real part (e), The i coefficient (f), The j coefficient (g), and The k coefficient (h) define q₂. Each is a required real number when its group is active. Enter 0 for a missing term – for example, g = 0 in the startup q₂. Larger coefficient magnitudes generally increase norms, but signs and multiplication order control the output components. A frequent mistake is to omit an implicit coefficient of 1 or – 1.

For rotation modes, Axis x, Axis y, and Axis z define any nonzero direction; the calculator normalizes it, so (1, 0, – 1) and (2, 0, – 2) describe the same axis. Rotation angle is a required signed real value, while Angle unit chooses degrees or radians and converts the current value when changed. The startup rotation example uses 60 degrees. For Rotate a 3D vector, Vector x, Vector y, and Vector z are also required; the saved demonstration is (1, 2, 3). Do not use a zero axis, because it has no direction.

Output guide

The header pills show Operation, First input norm, Second input norm, and Status. In quaternion modes, the first and second norms are ‖q₁‖ and ‖q₂‖; in rotation modes they map to the axis length and, when applicable, vector length. Primary result is the selected operation's main scalar, quaternion, matrix summary, or rotated vector. Secondary result appears when order creates a second meaningful answer, notably q₂ × q₁ and the alternative quotient. Output norm measures the magnitude of the primary quaternion or rotated vector, and Output scalar is the real coefficient when a quaternion is produced. Zero can be valid for a component or scalar; a zero norm is valid for the zero quaternion but prevents inversion or division.

The Component details table lists the real/i/j/k components for quaternion operations and adapts to axis, rotation-quaternion, or vector coordinates in rotation modes. Its values are estimates only in the sense of floating-point display rounding; algebraic identities are computed from full JavaScript number precision. Matrix representation is exact for the canonical floating-point values and appears for the 4 × 4 left-multiplication matrix or the 3 × 3 rotation matrix.

Worked example

With q₁ = 2 – i + 3j + k and q₂ = 5 – 4i + k, the real component of q₁ × q₂ is 2×5 – ( – 1)×( – 4) – 3×0 – 1×1 = 5. The i component is 2×( – 4) + ( – 1)×5 + 3×1 – 1×0 = – 10; the j component is 12; and the k component is 19. Therefore the first-open Primary result is 5 – 10i + 12j + 19k. Reversing the order gives 5 – 16i + 18j – 5k, demonstrating noncommutativity. The output norm, about 25.099801, also equals ‖q₁‖‖q₂‖ within floating-point precision.

Learn more

For rotations, Stanford's Quaternions and Rotations notes derive the algebra, axis-angle form, and vector conjugation method. NASA's Euler angles, quaternions, and transformation matrices report provides an engineering treatment of attitude representations and convention choices.