The quaternion calculator above performs the full set of quaternion operations on two inputs: addition, subtraction, both orders of the Hamilton product, division, conjugation, norm, inverse and normalisation. It then reads the result as a rotation, giving the axis and angle it represents and the equivalent Euler angles under a stated convention.
Arb Digital publishes it because quaternions are the standard way to represent three-dimensional rotation in graphics, robotics and aerospace, and because the arithmetic is easy to get subtly wrong by hand. The multiplication table alone has twelve sign-carrying products, and one flipped sign gives an answer that looks plausible and rotates the wrong way. Our complex number calculator handles the two-dimensional case, where multiplication is commutative and rotation is a single angle; quaternions are the four-dimensional extension where neither of those things is true.
What This Quaternion Calculator Does
It takes the four components of each quaternion, w plus xi plus yj plus zk, and computes every standard operation on them simultaneously. The headline panel shows whichever you select, but the supporting panels always display both products, both conjugates, both inverses and both norms, because the useful comparison is usually between two of those at once.
The rotation panel converts the result to axis-angle form and to Euler angles. Because only unit quaternions represent rotations, the conversion normalises first and reports the norm it divided out, so you can see whether the result was a rotation or merely a quaternion that happens to have four components.
The order panel is deliberately prominent. Quaternion multiplication is non-commutative, and the difference between the two orders is the single most common source of confusion for anyone coming from complex numbers or scalar algebra. Seeing both answers side by side makes the point better than any explanation.
How to Use It
- Enter the four components of each quaternion. The scalar part w comes first, followed by the i, j and k coefficients. Any real values are accepted.
- Choose the headline operation. The default is the Hamilton product in the order q₁ then q₂, which is the composition convention used in most graphics libraries.
- Compare the two orders in the panel below the grid. If they differ, the two rotations do not commute, which is true of almost every pair of rotations about different axes.
- Read the rotation panel for the axis, the angle in degrees and radians, and the Euler angles. The convention used is stated there explicitly.
- Use the example buttons to load the standard textbook integers, a clean ninety-degree rotation about the z axis, or two unit rotations whose composition is worth working through.
The Formula and How It Is Calculated
Quaternion multiplication follows from three rules Hamilton wrote on Broome Bridge in 1843: i squared, j squared and k squared all equal minus one, and their product ijk also equals minus one. Everything else follows, including the crucial fact that ij equals k while ji equals minus k. The MathWorld entry on quaternions sets out the algebra in full.
Written out, the product of (w₁, x₁, y₁, z₁) and (w₂, x₂, y₂, z₂) has scalar part w₁w₂ − x₁x₂ − y₁y₂ − z₁z₂, and vector part given by w₁ times the second vector, plus w₂ times the first vector, plus the cross product of the two vectors. That cross product is exactly where the non-commutativity comes from, because reversing the arguments flips its sign.
Worked example, using the integers loaded by default: q₁ is 1 + 2i + 3j + 4k and q₂ is 5 + 6i + 7j + 8k. The scalar part of the product is 5 minus 12 minus 21 minus 32, which is −60. The i component is 6 plus 10 plus 24 minus 28, which is 12. The j component is 7 minus 16 plus 15 plus 24, which is 30. The k component is 8 plus 14 minus 18 plus 20, which is 24. So the product is −60 + 12i + 30j + 24k. Reversing the order gives −60 + 20i + 14j + 32k: the scalar part is unchanged, which is always true, and the vector part differs by twice the cross product.
The conjugate negates the vector part and leaves w alone. The norm is the square root of the sum of the four squared components, so the norm of q₁ is the square root of 1 plus 4 plus 9 plus 16, which is the square root of 30, or 5.477226. The inverse is the conjugate divided by the squared norm, which for a unit quaternion is simply the conjugate.
Why Quaternions Beat Euler Angles for Rotation
A rotation in three dimensions has three degrees of freedom, so three Euler angles ought to be enough. They are not, and the reason is gimbal lock. At certain orientations two of the three axes align, one degree of freedom disappears, and the representation becomes singular: infinitely many angle triples describe the same orientation and small changes in orientation demand large jumps in the angles. Any system that interpolates or integrates orientations hits this eventually, and when it does the behaviour is not a small error but a visible flip.
A unit quaternion has four numbers subject to one constraint, so it also has three degrees of freedom, but it covers the rotation group without any singular points. Composition is one multiplication rather than a chain of trigonometric matrices. Interpolation between two orientations along the shortest path is a single formula. And renormalising after accumulated drift is a division by the norm, whereas re-orthogonalising a rotation matrix is a small optimisation problem.
The one genuine quirk is the double cover: q and −q describe exactly the same rotation, because the rotation formula applies the quaternion twice, once on each side, and the two sign changes cancel. This means quaternion space wraps around the rotation group twice, and a full turn of three hundred and sixty degrees returns you to −q rather than q. Interpolation code has to check the sign of the dot product and negate one input when it is negative, or the animation takes the long way round.
Rotating a Vector With a Quaternion
To rotate a vector v by a unit quaternion q, write v as a quaternion with zero scalar part and compute q times v times the conjugate of q. The result has zero scalar part again and its vector part is the rotated vector. The sandwiching is what makes the double cover harmless and what makes the operation a genuine rotation rather than a general linear map.
The relationship between the quaternion and the rotation it performs is that a rotation of angle θ about a unit axis (a₁, a₂, a₃) corresponds to the quaternion with scalar part cos(θ/2) and vector part sin(θ/2) times the axis. The half angle is not a typographical convenience: it is a consequence of the sandwiching, which applies the quaternion twice. This is why a quaternion with scalar part zero represents a rotation of one hundred and eighty degrees, not ninety. Our point rotation calculator handles the planar case directly, and the vector calculator covers the cross and dot products the formula is built from.
Euler Angle Conventions and Why They Disagree
There is no single definition of Euler angles. There are twelve valid axis sequences, each can be applied about fixed axes or about the axes as they move, and different fields have settled on different combinations. Aerospace generally uses yaw about z, then pitch about y, then roll about x, applied intrinsically. Classical mechanics texts often use the z-x-z sequence instead. Two correct tools using different conventions will report entirely different angle triples for the same rotation.
This page reports the intrinsic z-y-x sequence, sometimes written as yaw, pitch and roll in that order of application, which is the most common convention in robotics and flight dynamics. The pitch angle is clamped to the range from minus ninety to plus ninety degrees, and when it reaches either endpoint the yaw and roll become degenerate: only their sum or difference is determined. That is gimbal lock appearing in the conversion itself, and the tool says so when it happens. The MathWorld entry on Euler angles covers the alternative sequences.
Where Quaternions Are Used in Practice
Every major game engine stores object orientation as a quaternion internally, converting to a matrix only at the point of rendering. Spacecraft attitude determination and control systems use them because the algebra is compact enough for flight hardware and free of the singularities that would strand a control loop. Robot arm kinematics uses them for the same reason. Inertial measurement units in phones integrate gyroscope rates into a quaternion attitude estimate, renormalising each step.
They also appear in pure mathematics well beyond rotation: the unit quaternions form a group isomorphic to the special unitary group in two dimensions, which is the mathematical structure underlying spin in quantum mechanics. The double cover that looks like an inconvenience in graphics is, in that setting, the physically meaningful statement that a spin-one-half particle must be rotated through seven hundred and twenty degrees to return to its original state.
Arb Digital designs and builds interactive web tools, calculators and product experiences where the underlying logic has to be correct, not merely plausible.
See Our Web Design Work Talk To Our TeamCommon Mistakes to Avoid
- Assuming multiplication commutes — it does not, and the two orders correspond to composing rotations in opposite sequences. Check which order your library uses before porting code.
- Forgetting the half angle — the quaternion for a ninety-degree rotation has cosine of forty-five degrees as its scalar part, not cosine of ninety.
- Treating q and minus q as different rotations — they are the same rotation, and interpolation code must handle the sign or it will take the long path around.
- Using a non-unit quaternion to rotate — the sandwich product then scales the vector as well as rotating it, by the square of the norm.
- Comparing Euler angles across tools without checking the convention — there are twelve axis sequences and two application modes, so disagreement is usually a convention mismatch rather than an error.
Related Free Tools From Arb Digital
Work in two dimensions with the complex number calculator, handle dot and cross products with the vector calculator, rotate coordinates in the plane with the point rotation calculator, multiply the equivalent rotation matrices with the matrix calculator, or build higher-order products with the tensor product calculator. The full free online tools hub lists every algebra tool we publish.
Frequently Asked Questions
A four-component number of the form w plus xi plus yj plus zk, where i, j and k each square to minus one and their product ijk also equals minus one. It extends the complex numbers from two dimensions to four.
The vector part of the product contains a cross product of the two vector parts, and reversing the arguments flips the sign of a cross product. The scalar part is unaffected, which is why only the vector part differs.
Because a quaternion rotates a vector by sandwiching it between q and the conjugate of q, applying the quaternion twice. Each application contributes half the total angle.
Yes. The two sign changes in the sandwich product cancel exactly, so every rotation corresponds to two quaternions. This double cover is why interpolation code must check the sign before blending.
Gimbal lock is the loss of one degree of freedom when two Euler axes align, which makes the representation singular. Quaternions have no such singular orientations, though converting them to Euler angles reintroduces the problem.
Divide the conjugate by the squared norm. For a unit quaternion the squared norm is one, so the inverse is simply the conjugate, which is why unit quaternions are so cheap to work with.
The intrinsic z-y-x sequence, applied as yaw then pitch then roll, which is standard in robotics and flight dynamics. Other fields use different sequences and will report different angles for the same rotation.
Yes, but you must choose a side, because multiplication is non-commutative. This page computes the right division q₁ times the inverse of q₂, which is the more common convention.
This page explains a mathematical system for educational purposes. Rotation and Euler angle conventions vary between fields and software libraries, so check which convention your own system uses before transferring any result from this page into it.
