Editing Appendix/Mathematics/EulerAngles (section)

===Third Rotation===
As depicted in the right-most panel of Figure 2, rotate the triplet of unit vectors about the 3" axis by an angle, <math>\psi</math>.  The result is the desired black coordinate system labeled, <math>(\vec{e}_1, \vec{e}_2, \vec{e}_3)</math>.  Given that this step &#8212; like the first &#8212; calls for rotation about the "third" axis, we can immediately deduce that the relevant rotation matrix is,
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  <td align="right">
<math>\hat{R}_{3}(\psi)</math>
  </td>
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<math>~=</math>
  </td>
  <td align="left">
<math>
{\begin{bmatrix}
\cos\psi & \sin\psi & 0  \\
-\sin\psi & \cos\psi & 0 \\
0 & 0 & 1 
\end{bmatrix}} \, .
</math>
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