Editing Appendix/Mathematics/EulerAngles (section)

==Switching Coordinate Representations of a Vector==
Now, given that,
<div align="center">
<math>\vec{A}_\mathrm{body} = \hat{R} \cdot \vec{A}_{XYZ} \, ,</math>
</div>
the following three mapping relations must hold:

<table border="0" align="center" cellpadding="0">

<tr>
  <td align="right">
<math>A_1</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
A_X(\cos\psi \cos\phi - \sin\psi \sin\phi \cos\theta)
+ A_Y(\sin\phi \cos\psi + \sin\psi \cos\theta \cos\phi)
+ A_Z(\sin\psi \sin\theta) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>A_2</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
A_X(-\sin\psi \cos\phi - \sin\phi \cos\theta \cos\psi)
+ A_Y( - \sin\psi \sin\phi + \cos\psi \cos\theta\cos\phi )
+ A_Z (\sin\theta \cos\psi) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>A_3</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
A_X (\sin\theta\sin\phi)
+ A_Y (-\sin\theta \cos\phi)
+ A_Z (\cos\theta) \, .
</math>
  </td>
</tr>
</table>

Alternatively, given that,
<div align="center">
<math>\vec{A}_{XYZ} = \hat{R}^{-1} \cdot \vec{A}_\mathrm{body} \, ,</math>
</div>

the following additional three mapping relations must hold:

<table border="0" align="center" cellpadding="0">

<tr>
  <td align="right">
<math>A_X</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
A_1(\cos\psi \cos\phi - \sin\psi \sin\phi \cos\theta)
+ A_2(-\sin\psi \cos\phi - \sin\phi \cos\theta \cos\psi)
+ A_3(\sin\theta\sin\phi) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>A_Y</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
A_1(\sin\phi \cos\psi + \sin\psi \cos\theta \cos\phi)
+ A_2( - \sin\psi \sin\phi + \cos\psi \cos\theta\cos\phi )
+ A_3 (-\sin\theta \cos\phi) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>A_Z</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
A_1 (\sin\psi \sin\theta)
+ A_2 (\sin\theta \cos\psi)
+ A_3 (\cos\theta) \, .
</math>
  </td>
</tr>
</table>

----

Extending the [[#Simple_Numerical_Example|'''simple numerical example''' established above]], let's assume that the (red) vector that appears in both panels of Figure 1 has the ''inertial-frame'' (left panel) components,
<div align="center">
<math>(A_X, A_Y, A_Z) = (0.8, 0.8, 0.9) \, .</math>
</div>
The components of this same (red) vector as viewed from the ''body'' frame (right panel of Figure 1) should be,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>\begin{bmatrix}A_1 \\ A_2 \\ A_3 \end{bmatrix}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\hat{R} \cdot
\begin{bmatrix}A_X \\ A_Y \\ A_Z \end{bmatrix} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
{\begin{bmatrix}
0.7192 & 0.6861 & 0.1094  \\
-0.6619 & 0.6287 & 0.4082 \\
0.2113 & -0.3660 & 0.9063 
\end{bmatrix}}
\cdot
\begin{bmatrix}0.8 \\ 0.8 \\ 0.9 \end{bmatrix} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\begin{bmatrix} 1.2227 \\ 0.3408 \\ 0.6919 \end{bmatrix} \, .
</math>
  </td>
</tr>
</table>

Out of curiosity, what do we obtain if we use the inverse of the rotation matrix?  Well &hellip;
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>\begin{bmatrix}?_1 \\ ?_2 \\ ?_3 \end{bmatrix}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\hat{R}^{-1} \cdot
\begin{bmatrix}A_X \\ A_Y \\ A_Z \end{bmatrix} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
{\begin{bmatrix}
0.7192 & -0.6619 & 0.2113 \\
0.6861 & 0.6287 & -0.3660 \\
0.1094 & 0.4082 & 0.9063 
\end{bmatrix}} 
\cdot
\begin{bmatrix}0.8 \\ 0.8 \\ 0.9 \end{bmatrix} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\begin{bmatrix} 0.2360 \\ 0.7224 \\ 1.2298 \end{bmatrix} \, .
</math>
  </td>
</tr>
</table>