Free Quaternion Calculator

What Are Quaternions?

Quaternions are a four‑dimensional extension of complex numbers, first introduced by Sir William Hamilton to model three‑dimensional rotations. A quaternion is expressed as

$$q = a + b\,\mathbf{i} + c\,\mathbf{j} + d\,\mathbf{k},$$

where $a$, $b$, $c$, and $d$ are real numbers, and $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ are the fundamental unit quaternions. The real part is $a$, while the remaining three components form the “vector” part. When $c = d = 0$, the expression reduces to an ordinary complex number, highlighting how quaternions generalize the familiar two‑dimensional plane.

Adding and Subtracting Quaternions

Quaternion addition (and subtraction) works exactly like vector addition: you simply combine corresponding coefficients.

$$(a + b\mathbf{i} + c\mathbf{j} + d\mathbf{k}) + (e + f\mathbf{i} + g\mathbf{j} + h\mathbf{k}) = (a+e) + (b+f)\mathbf{i} + (c+g)\mathbf{j} + (d+h)\mathbf{k}.$$

This operation is associative, commutative, and every quaternion has an additive inverse. Subtraction follows the same principle – you add the negative of each coefficient.

Multiplying Quaternions – The Hamilton Product

Multiplication is more involved because it is non‑commutative. The rule relies on the products of the unit quaternions:

$$\mathbf{i}^2 = \mathbf{j}^2 = \mathbf{k}^2 = -1, \quad \mathbf{i}\mathbf{j} = \mathbf{k},\; \mathbf{j}\mathbf{k} = \mathbf{i},\; \mathbf{k}\mathbf{i} = \mathbf{j},\quad \mathbf{j}\mathbf{i} = -\mathbf{k},\; \mathbf{k}\mathbf{j} = -\mathbf{i},\; \mathbf{i}\mathbf{k} = -\mathbf{j}.$$

Using the distributive property, the product of two quaternions

$q_1 = a + b\mathbf{i} + c\mathbf{j} + d\mathbf{k}$ and $q_2 = e + f\mathbf{i} + g\mathbf{j} + h\mathbf{k}$

is called the Hamilton product. The result has the form:

$$\begin{aligned} q_1 q_2 = &\;(ae - bf - cg - dh) \\ &+ (af + be + ch - dg)\,\mathbf{i} \\ &+ (ag - bh + ce + df)\,\mathbf{j} \\ &+ (ah + bg - cf + de)\,\mathbf{k}. \end{aligned}$$

Because multiplication is not commutative, $q_1 q_2$ generally differs from $q_2 q_1$. The calculator therefore returns both orders when you perform a Quaternion Multiplication operation.

Norm, Conjugate, and Inverse

The norm (or magnitude) of a quaternion is analogous to the length of a four‑dimensional vector:

$$\|q\| = \sqrt{a^{2} + b^{2} + c^{2} + d^{2}}.$$

The conjugate flips the sign of the vector part:

$$\overline{q} = a - b\,\mathbf{i} - c\,\mathbf{j} - d\,\mathbf{k}.$$

Multiplying a quaternion by its conjugate yields the squared norm: $q\overline{q} = \|q\|^{2}$.

The inverse of a non‑zero quaternion is given by

$$q^{-1} = \frac{\overline{q}}{\|q\|^{2}}.$$

Because multiplication is non‑commutative, division is interpreted as multiplication by the inverse, and the order matters. The Quaternion Inverse Calculator functionality automatically computes both $q_1 q_2^{-1}$ and $q_2^{-1} q_1$.

Matrix Representation

A quaternion can be represented as a $4\times 4$ real matrix:

$$\begin{pmatrix} a & -b & -c & -d \\ b & a & -d & c \\ c & d & a & -b \\ d & -c & b & a \end{pmatrix}.$$

Alternatively, a $2\times 2$ complex matrix representation also exists. The norm equals the square root of the determinant, and the conjugate corresponds to the conjugate transpose.

Using Quaternions for 3D Rotation

A major application of quaternions is Quaternion Rotation in three‑dimensional space. Any rotation can be described by an axis (a unit vector $\mathbf{v}_a$) and an angle $\theta$. The rotation quaternion is

$$q = \cos\frac{\theta}{2} + \sin\frac{\theta}{2}\,(x_a\mathbf{i} + y_a\mathbf{j} + z_a\mathbf{k}),$$

where $\mathbf{v}_a = (x_a, y_a, z_a)$ is a unit vector. To rotate a point $\mathbf{v} = (x,y,z)$, first represent it as a pure quaternion $q_v = 0 + x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$. Then compute

$$q_v' = q \cdot q_v \cdot q^{-1}.$$

The resulting quaternion $q_v' = 0 + x'\mathbf{i} + y'\mathbf{j} + z'\mathbf{k}$ directly gives the rotated coordinates $(x', y', z')$. The same result can be obtained by converting the quaternion to a rotation matrix.

The Quaternion Rotation Calculator available in this tool handles all the intermediate steps: it converts an arbitrary axis vector to a unit vector, builds the rotation quaternion, and optionally outputs the equivalent rotation matrix.

Practical Example with the Calculator

Suppose you want to find the Hamilton product of

$q_1 = 2 - \mathbf{i} + 3\mathbf{j} + \mathbf{k}$ and $q_2 = 5 - 4\mathbf{i} + \mathbf{k}$.

Using the calculator’s “product” function, you enter the coefficients and obtain both possible products instantly.

Alternatively, to create a rotation quaternion for an axis defined by $\mathbf{v}_a = (1, 0, -1)$ with a $60^\circ$ angle, select “quaternion of rotation”, input the axis components and angle, and the tool returns the quaternion as well as the corresponding rotation matrix. This saves you from performing the manual trigonometric and algebraic steps.

The calculator supports all core operations: Add Quaternions, Quaternion Multiplication, Quaternion Norm Calculator, Quaternion Conjugate, Quaternion Inverse Calculator, and rotation‑related computations. Each operation delivers precise results and, when order matters, displays both possible outcomes.