Quaternion Calculator

Understanding Quaternions

Quaternions are a four-dimensional number system that extends the classical two-dimensional complex numbers. First discovered in 1843 by Irish mathematician Sir William Rowan Hamilton, quaternions provide an elegant, singularity-free algebraic framework for computing three-dimensional spatial orientations and rotations in robotics, 3D computer graphics, aerospace navigation, physics, and computer vision.

A quaternion $q$ is expressed in hypercomplex form with one real scalar part $w$ and three orthogonal imaginary vector parts $x, y, z$:

$$q = w + xi + yj + zk$$

Where $w, x, y, z \in \mathbb{R}$, and the imaginary unit elements $i, j, k$ obey Hamilton's fundamental non-commutative relations:

$$i^2 = j^2 = k^2 = ijk = -1$$

From these defining equations follow the cyclic multiplication rules:

$$ij = k, \quad jk = i, \quad ki = j$$ $$ji = -k, \quad kj = -i, \quad ik = -j$$

Quaternion Arithmetic Operations

Addition and Subtraction

Quaternion addition and subtraction are performed component-wise, analogous to 4D vector arithmetic:

$$q_1 \pm q_2 = (w_1 \pm w_2) + (x_1 \pm x_2)i + (y_1 \pm y_2)j + (z_1 \pm z_2)k$$

Hamilton Product (Multiplication)

Multiplication of two quaternions is associative and distributive, but strictly non-commutative ($q_1 q_2 \neq q_2 q_1$). Expanding the terms according to Hamilton's rules yields:

$$w = w_1 w_2 - x_1 x_2 - y_1 y_2 - z_1 z_2$$ $$x = w_1 x_2 + x_1 w_2 + y_1 z_2 - z_1 y_2$$ $$y = w_1 y_2 - x_1 z_2 + y_1 w_2 + z_1 x_2$$ $$z = w_1 z_2 + x_1 y_2 - y_1 x_2 + z_1 w_2$$

Norm, Conjugate, and Multiplicative Inverse

The conjugate of a quaternion $q^*$ reverses the sign of its imaginary vector components:

$$q^* = w - xi - yj - zk$$

The norm (or Euclidean magnitude) $|q|$ is the square root of the scalar product of $q$ with its conjugate:

$$|q|^2 = q q^* = w^2 + x^2 + y^2 + z^2, \quad |q| = \sqrt{w^2 + x^2 + y^2 + z^2}$$

For any non-zero quaternion, the multiplicative inverse $q^{-1}$ satisfies $q q^{-1} = q^{-1} q = 1$ and is calculated as:

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

Quaternion division $q_1 / q_2$ is defined by multiplying $q_1$ by the inverse of $q_2$: $q_1 \cdot q_2^{-1}$.

3D Spatial Rotations and Gimbal Lock Prevention

In 3D computer graphics, flight dynamics, and game engines (such as Unity and Unreal Engine), unit quaternions ($|q| = 1$, also known as versors) represent spatial rotations without the risk of gimbal lock encountered when using Euler angles (pitch, roll, and yaw).

To rotate a vector $\vec{v} = (v_x, v_y, v_z)$ by an angle $\theta$ around a unit axis $\vec{u} = (u_x, u_y, u_z)$, construct the unit quaternion:

$$q = \cos\left(\frac{\theta}{2}\right) + \vec{u} \sin\left(\frac{\theta}{2}\right)$$

The rotated vector $\vec{v}'$ is obtained by treating $\vec{v}$ as a pure quaternion $p = 0 + v_x i + v_y j + v_z k$ and evaluating the triple product:

$$p' = q \cdot p \cdot q^{-1} = q \cdot p \cdot q^*$$

Explore related mathematical utilities on OnlineMiniTools, including the Complex Calculator, Matrix Multiply Calculator, Matrix Inverse Calculator, and the Random Vector Generator.