Quaternion Calculator

A powerful tool for all quaternion operations.

Enter the components of your quaternions, select an operation, and get instant results. Supports addition, subtraction, multiplication, and more.

Examples

Explore these examples to understand common quaternion operations.

Adding Two Quaternions

add

Component-wise addition of two quaternions.

q1: w=1, x=2, y=3, z=4

q2: w=5, x=6, y=7, z=8

Multiplying Two Quaternions

multiply

Demonstrates the non-commutative multiplication.

q1: w=0, x=1, y=0, z=0

q2: w=0, x=0, y=1, z=0

Finding the Conjugate

conjugate

Negates the vector part of the quaternion.

q1: w=3, x=-1, y=2, z=5

Finding the Inverse

inverse

Calculates the inverse, useful for undoing rotations.

q1: w=1, x=1, y=1, z=1

Other Titles
Understanding Quaternions: A Comprehensive Guide
Dive deep into the world of quaternions, from basic definitions to their powerful applications in 3D graphics, robotics, and beyond.

What is a Quaternion?

  • An introduction to the 4D number system.
  • The relationship between quaternions, complex numbers, and 3D space.
  • The fundamental properties and notation.
A quaternion is a number system that extends the complex numbers. It was first introduced by Irish mathematician William Rowan Hamilton in 1843. Quaternions are expressed in the form q = w + xi + yj + zk, where w, x, y, and z are real numbers, and i, j, and k are the fundamental quaternion units.
Core Properties
The quaternion units follow these fundamental rules: i² = j² = k² = ijk = -1. This leads to other important identities, such as ij = k, jk = i, ki = j, and their anti-commutative counterparts ji = -k, kj = -i, ik = -j. This non-commutative property (ij ≠ ji) is a key feature of quaternions and is crucial for representing 3D rotations correctly.
A quaternion can be seen as a pair of a scalar part (w) and a vector part (v = xi + yj + zk). This structure makes them incredibly useful for representing orientations and rotations in three-dimensional space.

Key Quaternion Identities

  • ij = k, but ji = -k (Non-commutativity)
  • i² = -1 (Similar to complex numbers)
  • A pure quaternion has a zero scalar part (w=0).

Step-by-Step Guide to Using the Quaternion Calculator

  • How to input quaternions for calculations.
  • A breakdown of each available operation.
  • Interpreting the results for different operations.
Our calculator simplifies performing complex quaternion arithmetic. Follow these steps to get started.
1. Select an Operation
Use the dropdown menu to choose the desired operation. Note that 'Conjugate', 'Norm', 'Inverse', and 'Normalize' are unary operations and only require the first quaternion (q1) to be filled.
2. Input Quaternion Components
For each quaternion, enter its four components (w, x, y, z) into the respective input fields. The calculator accepts integers and decimal values.
3. Calculate and View Results
Click the 'Calculate' button to perform the operation. The result will be displayed below. For operations that yield a quaternion, all four components will be shown. For the 'Norm' operation, a single scalar value will be shown.

Operation Walkthrough

  • For Addition: Fill in all fields for q1 and q2.
  • For Conjugate: Fill in only the fields for q1.
  • The calculator automatically handles whether q2 is needed.

Real-World Applications of Quaternions

  • Why quaternions are essential in computer graphics.
  • Their role in robotics and aerospace engineering.
  • Applications in physics and virtual reality.
Quaternions are not just a mathematical curiosity; they are a cornerstone of modern technology.
Computer Graphics and Game Development
The most famous application of quaternions is in representing 3D rotations. They avoid the problem of 'gimbal lock,' which can plague other rotation representations like Euler angles. Quaternions also allow for smooth and efficient interpolation between orientations using algorithms like SLERP (Spherical Linear Interpolation), which is vital for animation.
Robotics and Drones
In robotics, determining and controlling the orientation of a robot arm, a vehicle, or a drone is critical. Quaternions provide a robust and computationally efficient way to track and manipulate this orientation in real-time.
Aerospace and Satellite Navigation
The attitude (orientation) of spacecraft and satellites must be precisely controlled. Quaternions are used in the attitude control systems of these vehicles to ensure they are pointing in the correct direction.

Industry Examples

  • Animating a character's joints in a video game.
  • Controlling the flight orientation of a quadcopter.
  • Tracking the orientation of a virtual reality headset.

Common Misconceptions and Correct Methods

  • Dispelling the myth of quaternion commutativity.
  • Understanding the difference between unit quaternions and general quaternions.
  • Clarifying how quaternions encode rotation.
Misconception 1: Quaternion Multiplication is Commutative
Incorrect Assumption: Many people initially think that q1 * q2 is the same as q2 * q1. Correction: This is false. The order of multiplication matters immensely. In terms of rotations, q1 * q2 means applying rotation q2 first, followed by rotation q1. Reversing the order produces a different final orientation.
Misconception 2: Any Quaternion Represents a Pure Rotation
Incorrect Assumption: Any set of four numbers forming a quaternion is a 3D rotation. Correction: Only unit quaternions (quaternions with a norm/magnitude of 1) represent pure 3D rotations. A non-unit quaternion represents a combination of rotation and scaling. This is why normalization is a critical operation.
Misconception 3: The w Component is the Rotation Angle
Incorrect Assumption: The w component directly represents an angle. Correction: The rotation is encoded in all four components. For a rotation of angle θ around a unit vector axis (ax, ay, az), the corresponding quaternion is q = cos(θ/2) + (ax*sin(θ/2))i + (ay*sin(θ/2))j + (az*sin(θ/2))k.

Mathematical Derivations and Formulas

  • The formula for quaternion multiplication.
  • Derivation of the quaternion inverse.
  • How a quaternion rotation is applied to a vector.
Quaternion Multiplication
Given q1 = w1 + x1i + y1j + z1k and q2 = w2 + x2i + y2j + z2k, their product q1 * q2 is: w = w1w2 - x1x2 - y1y2 - z1z2, x = w1x2 + x1w2 + y1z2 - z1y2, y = w1y2 - x1z2 + y1w2 + z1x2, z = w1z2 + x1y2 - y1x2 + z1w2.
Quaternion Inverse
The inverse of a quaternion q is defined as q⁻¹ = q* / ||q||², where q* is the conjugate of q and ||q|| is its norm. The conjugate q* is w - xi - yj - zk, and the squared norm ||q||² is w² + x² + y² + z².
Rotating a Vector
To rotate a vector v using a unit quaternion q, you first represent the vector as a pure quaternion p = 0 + v_x i + v_y j + v_z k. The rotated vector v' is then found by computing p' = q * p * q⁻¹, where q⁻¹ is the inverse of q (which is also its conjugate, since q is a unit quaternion). The vector part of the resulting quaternion p' is the rotated vector v'.