Editing Appendix/Mathematics/EulerAngles (section)

===First Rotation===
As depicted in the left-most panel of Figure 2, rotate the triplet of unit vectors about the <math>Z</math> (''i.e.,'' <math>\vec{e}_Z</math>) axis by an angle, <math>\phi</math>. The result is the ''green'' coordinate system labeled, <math>(1', 2', 3')</math>.  Note: (a) The 3' axis remains aligned with the inertial-frame Z-axis; and (b) we will refer to the 1' axis as the ''line of nodes''.  Given that the 3' axis is aligned with the Z-axis &#8212; that is, <math>\vec{e}_{3'} = \vec{e}_Z</math> &#8212; it is easy to recognize that the other two mappings are:  
<div align="center"><math>\vec{e}_{1'} = \vec{e}_X \cos\phi + \vec{e}_Y \sin\phi \, ,</math> &nbsp; &nbsp; and &nbsp; &nbsp; <math>\vec{e}_{2'} = \vec{e}_Y \cos\phi - \vec{e}_X \sin\phi</math>.</div>
Hence, the corresponding rotation matrix is,
<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right">
<math>\hat{R}_{3}(\phi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
{\begin{bmatrix}
\vec{e}_{1'} \cdot\vec{e}_X & \vec{e}_{1'} \cdot\vec{e}_Y & \vec{e}_{1'} \cdot\vec{e}_Z  \\
\vec{e}_{2'} \cdot\vec{e}_X & \vec{e}_{2'} \cdot\vec{e}_Y & \vec{e}_{2'} \cdot\vec{e}_Z \\
\vec{e}_{3'} \cdot\vec{e}_X & \vec{e}_{3'} \cdot\vec{e}_Y & \vec{e}_{3'} \cdot\vec{e}_Z 
\end{bmatrix}}
=
{\begin{bmatrix}
\cos\phi & \sin\phi & 0  \\
-\sin\phi & \cos\phi & 0 \\
0 & 0 & 1 
\end{bmatrix}} \, .
</math>
  </td>
</tr>
</table>
Note that the subscript, 3, has been attached to <math>\hat{R}</math> in order to indicate that rotation was about the "third" axis.