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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:

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

## See also

Last update : 05/14/2021