Compute Quaternion Magnitude

Calculator and formula to compute the length (magnitude) of a quaternion

Quaternion Magnitude Calculator

Quaternion Norm (4D length)

Computes the length (magnitude) of a quaternion using 4D Pythagoras: |q| = √(w² + x² + y² + z²)

Select calculation
|q| = √(w² + x² + y² + z²)
|q|² = w² + x² + y² + z²
Enter quaternion components
Rotation component
X vector component
Y vector component
Z vector component
Quaternion magnitude result
Magnitude |q|:
4D length of the quaternion
Calculation: |q| = √(w² + x² + y² + z²)

Quaternion Magnitude Info

Magnitude properties

Norm: Always non-negative

|q| ≥ 0 4D Pythagoras Euclidean norm

Unit quaternion: |q| = 1 for rotations
Normalization: q_norm = q / |q|

Examples
|[1, 0, 0, 0]| = 1 (unit quaternion)
|[1, 1, 1, 1]| = √4 = 2
|[0, 3, 4, 0]| = √25 = 5


Formulas for Quaternion Magnitude

Quaternion magnitude (norm)
\[|q| = \sqrt{w^2 + x^2 + y^2 + z^2}\]

4D Pythagoras for quaternions

Squared magnitude
\[|q|^2 = w^2 + x^2 + y^2 + z^2\]

Without square root (more efficient)

Conjugate representation
\[|q|^2 = q \cdot \overline{q}\]

Quaternion times its conjugate

Normalization
\[\hat{q} = \frac{q}{|q|} = \frac{q}{\sqrt{w^2 + x^2 + y^2 + z^2}}\]

Create a unit quaternion

Examples for Quaternion Magnitude

Example 1: Simple magnitude
q = 1 + 3i + 5j + 2k
Step 1: Square components \[\begin{align} w^2 &= 1^2 = 1 \\ x^2 &= 3^2 = 9 \\ y^2 &= 5^2 = 25 \\ z^2 &= 2^2 = 4 \end{align}\] Step 2: Sum and root \[|q| = \sqrt{1 + 9 + 25 + 4} = \sqrt{39} \approx 6.24\]

Magnitude: √39 ≈ 6.24

Example 2: Unit quaternion
q = 0.5 + 0.5i + 0.5j + 0.5k
\[\begin{align} |q|^2 &= (0.5)^2 + (0.5)^2 + (0.5)^2 + (0.5)^2 \\ &= 0.25 + 0.25 + 0.25 + 0.25 = 1 \\ |q| &= \sqrt{1} = 1 \end{align}\]

Unit quaternion: |q| = 1

Geometric meaning
4D distance
To the origin
Rotation strength
For unit quaternions
Normalization
For valid rotations
Scaling
Quaternion magnitude

For 3D rotations quaternions must be normalized (|q| = 1)

Step-by-step guide
Compute magnitude
  1. Square all four components w, x, y, z
  2. Add the squares: w² + x² + y² + z²
  3. Take the square root of the sum
Squared magnitude
  1. Square all four components
  2. Add the squares: w² + x² + y² + z²
  3. Done (no square root)

Applications of the quaternion magnitude

The quaternion magnitude is fundamental for many 3D applications:

Computer Graphics & Animation
  • Normalization for valid rotations
  • Check for unit quaternions
  • Compute interpolation weights
  • Control quaternion scaling
Robotics & Control
  • Check orientation stability
  • Sensor calibration and validation
  • Quaternion filtering for noise suppression
  • Compute rotational speeds
Aviation & Navigation
  • Attitude control
  • Gyroscope data processing
  • Drone stabilization
  • Inertial navigation systems
Mathematics & Physics
  • 4D geometry and topology
  • Spinor calculations
  • Symmetry operations
  • Lorentz group computations

Quaternion magnitude: the 4D norm

The quaternion magnitude is the direct generalization of the Euclidean norm to the four-dimensional space of quaternions. It measures the "length" of a quaternion in 4D and is essential for normalization to unit quaternions that represent valid 3D rotations. The magnitude connects algebraic quaternion structure with the geometric interpretation as a rotation operator and underlies stable numerical algorithms in 3D graphics and robotics.

Summary

The quaternion magnitude extends Pythagoras to four dimensions and links quaternion algebra with 3D geometry. Normalizing by the magnitude yields unit quaternions — the numerically stable representation of 3D rotations. From games to robotics, the quaternion magnitude is a key tool for precise rotation computations.




More Quaternion Functions

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