Quaternion Subtraction

Component-wise subtraction of two quaternions

Quaternion Subtraction Calculator

Quaternion Subtraction

Subtracts two quaternions q₁ - q₂ by component-wise difference of all four components (W, X, Y, Z)

Subtraction properties

Formula: q₁ - q₂ = (w₁-w₂) + (x₁-x₂)i + (y₁-y₂)j + (z₁-z₂)k
Not commutative: q₁ - q₂ ≠ q₂ - q₁ (order matters!)
Use: compute differences, relative orientations

Enter two quaternions to subtract
Minuend (q₁)
Subtrahend (q₂)
q₁ - q₂: Minuend minus subtrahend
Not commutative: q₁ - q₂ ≠ q₂ - q₁
Quaternion subtraction result
W (scalar):
X (i comp.):
Y (j comp.):
Z (k comp.):
Subtraction: q₁ - q₂ = (w₁-w₂, x₁-x₂, y₁-y₂, z₁-z₂)

Subtraction info

Properties

Not commutative: q₁ - q₂ ≠ q₂ - q₁
Associative: (q₁ - q₂) - q₃ = q₁ - (q₂ + q₃)
Distributive: over scalar multiplication

Not commutative Component-wise Difference

Order: q₁ - q₂ ≠ q₂ - q₁
Simple: component by component

Special cases
q - q = 0: zero quaternion
q - 0 = q: quaternion unchanged
0 - q = -q: negated quaternion
q₁ - q₂ = q₁ + (-q₂): addition of negation


Formulas for quaternion subtraction

General subtraction 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 difference

Component-wise representation
\[\begin{align} w &= w_1 - w_2 \\ x &= x_1 - x_2 \\ y &= y_1 - y_2 \\ z &= z_1 - z_2 \end{align}\]

Each component subtracted individually

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

As a 4D vector subtraction

Relation to addition
\[q_1 - q_2 = q_1 + (-q_2)\]

Subtraction as addition of negation

Algebraic properties
\[\begin{align} q_1 - q_2 &\neq q_2 - q_1 \\ (q_1 - q_2) - q_3 &= q_1 - (q_2 + q_3) \\ \lambda(q_1 - q_2) &= \lambda q_1 - \lambda q_2 \end{align}\]

Not commutative, but distributive over scalars

Examples for quaternion subtraction

Example 1: Basic calculation
q₁ = 3 + 2i + 4j + 1k q₂ = 1 + 3i + 5j + 2k
Subtraction q₁ - q₂: \[\begin{align} w &= 3 - 1 = 2 \\ x &= 2 - 3 = -1 \\ y &= 4 - 5 = -1 \\ z &= 1 - 2 = -1 \end{align}\]

q₁ - q₂ = 2 - 1i - 1j - 1k

Example 2: Non-commutativity
q₁ = 5 + 3i + 2j + 1k q₂ = 2 + 1i + 1j + 3k
q₁ - q₂: \[= (5-2) + (3-1)i + (2-1)j + (1-3)k = 3 + 2i + 1j - 2k\] q₂ - q₁: \[= (2-5) + (1-3)i + (1-2)j + (3-1)k = -3 - 2i - 1j + 2k\]

q₁ - q₂ ≠ q₂ - q₁

Example 3: Self-subtraction
q = 4 + 3i + 2j + 1k
q - q: \[\begin{align} w &= 4 - 4 = 0 \\ x &= 3 - 3 = 0 \\ y &= 2 - 2 = 0 \\ z &= 1 - 1 = 0 \end{align}\] Result: Zero quaternion

q - q = 0 + 0i + 0j + 0k

Example 4: Subtraction from zero
q = 2 + 1i + 3j + 4k 0 = 0 + 0i + 0j + 0k
q - 0: \[= 2 + 1i + 3j + 4k = q\] 0 - q: \[= 0 - 2 + (0-1)i + (0-3)j + (0-4)k = -q\]

q - 0 = q, but 0 - q = -q

Geometric meaning
Difference vector
Direction & magnitude
Relative position
To the reference point
Orientation difference
Between rotations
Rate of change
Discrete derivative

Subtraction produces a "difference quaternion" that describes the deviation between two quaternions

Step-by-step guide
Preparation
  1. Write both quaternions in standard form
  2. Decide order: q₁ - q₂ (minuend - subtrahend)
  3. Identify all four components
  4. Systematically subtract component-wise
Execution
  1. W component: w₁ - w₂
  2. X component: x₁ - x₂
  3. Y component: y₁ - y₂
  4. Z component: z₁ - z₂

Applications of quaternion subtraction

Quaternion subtraction is important for difference analysis and relative calculations:

3D Graphics & Animation
  • Orientation difference: between two rotation states
  • Animation interpolation: start & end difference
  • Camera movement: relative position change
  • Object tracking: motion vectors
Robotics & Control
  • Setpoint error: compute control deviation
  • Path correction: deviation from planned route
  • Sensor calibration: determine offsets
  • Motion analysis: velocity approximation
Numerical analysis
  • Discrete derivative: approximate change rates
  • Error computation: quantify deviations
  • Residual analysis: equation system solutions
  • Convergence test: iterative methods
Key properties
  • Not commutative: order matters
  • Simplicity: component-wise subtraction
  • Linearity: distributive over scalars
  • Inverse operation: to addition

Quaternion subtraction: differences in 4D space

Quaternion subtraction is a fundamental operation that computes the difference between two points in 4D quaternion space. While conceptually simple — just component-wise subtraction — it has important applications in analyzing orientation changes and relative positions. The non-commutative nature (q₁ - q₂ ≠ q₂ - q₁) reflects the direction dependence of differences. In 3D graphics subtraction is often used to quantify the change between two rotation states or to compute motion vectors for animations.

Summary

Quaternion subtraction combines algebraic simplicity with practical versatility. As a component-wise operation it is easy to understand and implement, but its applications range from foundational research to complex control systems. The non-commutative property makes it the natural choice for computing directed differences, while its relation to addition (q₁ - q₂ = q₁ + (-q₂)) embeds it comfortably within larger algebraic structures. Modern applications in robotics, animation and simulation use subtraction for real-time computations of orientation differences and motion analysis.




More Quaternion Functions

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