Calculadora de Quaternion (norma e conjugado)
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
|q| = √(a² + b² + c² + d²) ; q̄ = a − bi − cj − dk
About this calculator
The Quaternion Calculator allows you to input a quaternion in the form q = a + bi + cj + dk and obtain its norm (magnitude) and its conjugate. Quaternions are extensions of complex numbers, with three imaginary components i, j, k, used in computer graphics, robotics, and physics to represent rotations and orientations in three-dimensional space.
The norm is calculated using the formula |q| = √(a² + b² + c² + d²), which measures the length of the quaternion in four-dimensional space. The conjugate is given by q̄ = a − bi − cj − dk, which flips the sign of the imaginary parts. These operations are essential for normalizing quaternions (to obtain a versor) and for computing inverses.
Use this tool when working with 3D animations, physics simulations, or inertial navigation, where quaternions avoid issues like gimbal lock. Simply fill in the fields a, b, c, d with real numbers and click calculate. The result displays both values simultaneously.
Caution: ensure that the real part (a) and imaginary parts (b, c, d) are real numbers. The norm will always be a non-negative real number. The conjugate is used to compute the inverse of a unit quaternion (inverse = conjugate for norm = 1).
Frequently asked questions
What are quaternions and what are they used for?
Quaternions are hypercomplex numbers used to represent rotations in 3D space without gimbal lock issues. They are widely used in computer graphics, robotics, and games.
How do you calculate the norm of a quaternion?
The norm is the square root of the sum of squares of all components: |q| = √(a² + b² + c² + d²). It represents the length of the quaternion.
What is the conjugate of a quaternion?
The conjugate of q = a + bi + cj + dk is q̄ = a − bi − cj − dk. It flips the sign of the imaginary parts.
What is the difference between norm and magnitude?
In quaternions, norm and magnitude are synonymous: both refer to the value |q| computed by the formula above.
Can I use negative numbers in the fields?
Yes, the components a, b, c, d can be any real number, positive or negative.