## Apply rotation in three-dimensional space through complex vectors

Quaternions are vectors used for computing rotations in mechanics, aerospace, computer graphics, vision processing, and other applications. They consist of four elements: three that extend the commonly known imaginary number and one that defines the magnitude of rotation. Quaternions are commonly denoted as:

$q=w+x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\quad\text{where}\quad \mathbf{i}^2=\mathbf{j}^2=\mathbf{k}^2=\mathbf{i}\mathbf{j}\mathbf{k}=-1$

This rotation format requires less computation than a rotation matrix.

Common tasks for using quaternion include:

• Converting between quaternions, rotation matrices, and direction cosine matrices
• Performing quaternion math such as norm inverse and rotation
• Simulating premade six degree-of freedom (6DoF) models built with quaternion math

For details, see MATLAB® and Simulink® that enable you to use quaternions without a deep understanding of the mathematics involved.

### Software Reference

See also: Euler angles, linearization, numerical analysis, design optimization, real-time simulation, Monte Carlo simulation, model-based testing, Aerospace Toolbox, Aerospace Blockset, Sensor Fusion and Tracking Toolbox