Quaternion Division

Calculator and formula for quaternion division

Quaternion Division Calculator

Quaternion Division

Divides a quaternion q₁ by a second quaternion q₂ by multiplying with the inverse: q₁ ÷ q₂ = q₁ × q₂⁻¹

Quaternion division via inverse

Division is performed by multiplying with the conjugated quaternion and dividing by the norm squared: q₁ ÷ q₂ = (q₁ × q₂*) / |q₂|²

Enter two quaternions for division
Dividend (q₁)
Divisor (q₂)
Quaternion Division Result
W (scalar):
X (i comp.):
Y (j comp.):
Z (k comp.):
Quaternion division: q₁ ÷ q₂ = q₁ × q₂⁻¹ = (q₁ × q₂*) / |q₂|²

Quaternion Division Info

Division properties

Not commutative: q₁ ÷ q₂ ≠ q₂ ÷ q₁

Not commutative Complex Inverse required

Warning: Division by a zero quaternion is impossible
Complex: Computed via conjugation and norm

Division procedure
1. Conjugate the divisor (q₂*)
2. Compute norm squared (|q₂|²)
3. Multiply: q₁ × q₂*
4. Divide by |q₂|²


Formulas for Quaternion Division

General formula
\[q_1 \div q_2 = q_1 \times q_2^{-1} = \frac{q_1 \times q_2^*}{|q_2|^2}\]

Division by multiplication with the inverse

Conjugated quaternion
\[q_2^* = w_2 - x_2i - y_2j - z_2k\]

Sign flip of the vector components

Norm squared
\[|q_2|^2 = w_2^2 + x_2^2 + y_2^2 + z_2^2\]

Sum of squared components

Detailed calculation
\[\begin{align} q_1 \div q_2 &= \frac{(w_1 + x_1i + y_1j + z_1k) \times (w_2 - x_2i - y_2j - z_2k)}{w_2^2 + x_2^2 + y_2^2 + z_2^2} \end{align}\]

Full formula with all components

Examples for Quaternion Division

Example 1: Simple division
q₁ = 3 + 5i + 4j + 8k q₂ = 2 + 3i + 5j + 2k
Step 1: Conjugate of q₂ \[q_2^* = 2 - 3i - 5j - 2k\] Step 2: Norm squared \[|q_2|^2 = 4 + 9 + 25 + 4 = 42\] Step 3: Multiplication \[q_1 \times q_2^*\] Step 4: Divide by 42

Complex calculation required

Example 2: Division by a pure quaternion
q₁ = 1 + 0i + 0j + 0k q₂ = 0 + 1i + 0j + 0k
\[\begin{align} q_2^* &= 0 - 1i - 0j - 0k = -i \\ |q_2|^2 &= 0 + 1 + 0 + 0 = 1 \\ q_1 \times q_2^* &= 1 \times (-i) = -i \\ \text{Result} &= \frac{-i}{1} = -i \end{align}\]

Result: 0 - 1i + 0j + 0k

Geometric meaning
Rotation inversion
Inverse rotation
Orientation difference
Relative alignment
Transformation
Coordinate change
Calibration
Offset correction

Division corresponds to inverting a rotation or computing the relative orientation between two states

Step-by-step guide
Preparation
  1. Write both quaternions in standard form
  2. Check: divisor ≠ 0 (|q₂| ≠ 0)
  3. Form conjugate of divisor: q₂*
  4. Compute norm squared of divisor: |q₂|²
Execution
  1. Quaternion multiplication: q₁ × q₂*
  2. All components of the result
  3. Divide all components by |q₂|²
  4. Simplify and format

Applications of Quaternion Division

Quaternion division has important practical applications:

3D Graphics & Animation
  • Relative rotations between objects
  • Inverse kinematics: compute joint angles
  • Camera orientation: reverse view direction
  • Coordinate transforms between systems
Robotics & Navigation
  • Sensor calibration: offset correction
  • Path planning: reversed motions
  • Orientation control: desired-actual difference
  • Gimbal-lock-free rotation
Mathematics & Physics
  • Quantum mechanics: spinor operations
  • Crystallography: symmetry operations
  • Signal processing: filter design
  • Numerical solution of differential equations
Important notes
  • Not commutative: q₁/q₂ ≠ q₂/q₁
  • Division by zero impossible
  • Numerical stability for small norms must be considered
  • Often slower than multiplication with inverse

Quaternion Division: complex but powerful operation

Quaternion division is one of the more complex operations in quaternion algebra and is performed by multiplying with the inverse quaternion. Unlike simple addition, it requires computing the conjugate quaternion and the norm squared. Division is not commutative and geometrically corresponds to inverting a rotation or computing the relative orientation between two spatial states. In practice it is used for inverse transformations, sensor calibration and orientation problems in 3D graphics and robotics.

Summary

Quaternion division is mathematically demanding and requires careful computation of the inverse quaternion via conjugation and norm squared. Although more complex than addition or multiplication, it is indispensable for applications that require inverse rotations or relative orientations. The geometric meaning corresponds to "undoing" a rotation or answering the question: "Which rotation takes state A to state B?" Special care is needed in numerical computations since small divisors can lead to instabilities.




More Quaternion Functions

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