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# Quaternion conjugate

## Introduction

The conjugate can be used to swap the relative frames described by an orientation. For example, if $${}^BQ_A$$ describes the orientation of frame $$B$$ relative to frame $$A$$, the conjugate of $$\overline { {}^BQ_A }={}^AQ_B$$ describes the orientation of frame $$A$$ relative to frame $$B$$. Relative orientations can also be seen as transformation from one frame to another.

## Notation

The quaternion conjugate is denoted by one of the following notation : $$Q^*$$ , $$\overline Q$$ or $$Q^T$$. The prefered notation is $$\overline Q$$.

## Properties

Considere the quaternion $$Q$$ defined by:

$$Q = \left[ \begin{matrix} a && b && c && d \end{matrix} \right]$$

The conjugate of $$Q$$ is given by:

$$\overline Q = \left[ \begin{matrix} a && -b && -c && -d \end{matrix} \right]$$