Quaternion Addition

Calculator and formula for component-wise addition of quaternions

Quaternion Addition Calculator

Quaternion Addition

Adds two quaternions q₁ + q₂ by component-wise addition of all four components (w, x, y, z)

Component-wise addition of quaternions

Quaternion addition is performed by adding corresponding components: (w₁+w₂) + (x₁+x₂)i + (y₁+y₂)j + (z₁+z₂)k

Enter two quaternions
Quaternion 1 (q₁)
Quaternion 2 (q₂)
Quaternion Addition Result
W (scalar):
X (i comp.):
Y (j comp.):
Z (k comp.):
Quaternion addition: q₁ + q₂ = (w₁+w₂) + (x₁+x₂)i + (y₁+y₂)j + (z₁+z₂)k

Quaternion Addition Info

Addition Properties

Commutative: q₁ + q₂ = q₂ + q₁

Commutative Associative Component-wise

Simple: Add corresponding components
Fundamental: Basic operation for quaternions

Quaternion Components
W: Scalar part (rotation component)
X, Y, Z: Vector components (i, j, k)


Formulas for Quaternion Addition

General formula
\[q_1 + q_2 = (w_1 + w_2) + (x_1 + x_2)i + (y_1 + y_2)j + (z_1 + z_2)k\]

Component-wise addition of all four parts

Vector representation
\[\vec{q_1} + \vec{q_2} = \begin{pmatrix} w_1 + w_2 \\ x_1 + x_2 \\ y_1 + y_2 \\ z_1 + z_2 \end{pmatrix}\]

As a 4D vector addition

Individual components
\[\begin{align} w &= w_1 + w_2 \\ x &= x_1 + x_2 \\ y &= y_1 + y_2 \\ z &= z_1 + z_2 \end{align}\]

Each component separately

Properties
\[\begin{align} q_1 + q_2 &= q_2 + q_1 \\ (q_1 + q_2) + q_3 &= q_1 + (q_2 + q_3) \\ q + 0 &= q \end{align}\]

Commutative, associative, neutral element

Examples for Quaternion Addition

Example 1: Simple addition
q₁ = 3 + 2i + 4j + 1k q₂ = 1 + 3i + 5j + 2k
Step 1: W components \[w = 3 + 1 = 4\] Step 2: X components \[x = 2 + 3 = 5\] Step 3: Y components \[y = 4 + 5 = 9\] Step 4: Z components \[z = 1 + 2 = 3\]

Result: q = 4 + 5i + 9j + 3k

Example 2: With negative numbers
q₁ = 2 - 1i + 3j - 2k q₂ = -1 + 4i - 1j + 5k
\[\begin{align} w &= 2 + (-1) = 1 \\ x &= (-1) + 4 = 3 \\ y &= 3 + (-1) = 2 \\ z &= (-2) + 5 = 3 \end{align}\]

Result: q = 1 + 3i + 2j + 3k

Geometric interpretation
Rotation composition
Not the same as rotation!
Linear combination
Weighted averaging
Interpolation
Between states
Superposition
Additive combination

Addition is not rotation composition — use multiplication for that

Step-by-step guide
Preparation
  1. Write both quaternions in the form w + xi + yj + zk
  2. Identify components clearly
  3. Treat missing components as 0
Execution
  1. Add W components: w₁ + w₂
  2. Add X components: x₁ + x₂
  3. Add Y components: y₁ + y₂
  4. Add Z components: z₁ + z₂

Applications of Quaternion Addition

Quaternion addition has various practical applications:

Computer Graphics & Animation
  • Interpolation between rotation states
  • Weighted combinations of orientations
  • Skeletal animation: bone blending
  • Morphing between 3D object orientations
Robotics & Control
  • Controller integration: joystick inputs
  • Sensor fusion: combining orientation data
  • Path planning: intermediate states
  • Calibration: offset corrections
Mathematics & Simulation
  • Numerical integration of rotation differences
  • Finite element methods with orientations
  • Statistical evaluation of orientation data
  • Quaternion-based filtering methods
Important limitation
  • Not for rotation composition!
  • Addition ≠ sequential application of rotations
  • For rotations use quaternion multiplication
  • Addition: weighting, interpolation, superposition

Quaternion Addition: Simple linear combination

Quaternion addition is the most elementary operation in quaternion algebra and is performed by component-wise addition. Unlike quaternion multiplication, which is used for rotation composition, addition serves linear combination and interpolation. It is commutative and associative, forms an abelian group, and allows weighted averages between quaternion states. In practice it is used in sensor fusion, animation of orientation transitions and numerical integration of orientation changes.

Summary

Quaternion addition is mathematically simple—just component-wise addition—but conceptually important: it is not the geometric composition of rotations, but the algebraic linear combination in 4D. While multiplication chains rotations, addition combines quaternion objects. This distinction is crucial for correct application in 3D graphics, robotics and simulation, where both operations have their roles.




More Quaternion Functions

Addition  •  Subtraction  •  Division  •  Multiplication  •  Concatenate  •  Length  •  Interpolation  •  Normalize  •  Scalar Multiplication  •  Dot Product  •  Yaw-Pitch-Roll  •  Conjugates  •  Inverse  •  Negation  •