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:<math>\mathbf{q}_3 = \sin(\alpha/2)\cos(\beta_z)</math>
where α is a simple rotation angle (the value in radians of the angle of rotation) and cos(β<sub>''x''</sub>), cos(β<sub>''y''</sub>) and cos(β<sub>''z''</sub>) are the "[[Unit vector|direction cosines]]" locating the axis of rotation (Euler's Theorem).
===Relationship with Tait-Bryan angles===▼
[[Image:Flight dynamics with text.png|right|thumb]]▼
Similarly for Euler angles, we use the [[Tait-Bryan angles]] (in terms of [[flight dynamics]]):▼
* Roll - <math>\phi</math>: rotation about the X-axis▼
* Pitch - <math>\theta</math>: rotation about the Y-axis▼
* Yaw - <math>\psi</math>: rotation about the Z-axis▼
where the X-axis points forward, Y-axis to the right and Z-axis downward and in the example to follow the rotation occurs in the order yaw, pitch, roll (about body-fixed axes).▼
== Rotation matrices ==
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-\sin\theta & \sin\phi \cos\theta & \cos\phi \cos\theta \\
\end{bmatrix}</math>
▲===Relationship with Tait-Bryan angles===
▲[[Image:Flight dynamics with text.png|right|thumb]]
▲Similarly for Euler angles, we use the [[Tait-Bryan angles]] (in terms of [[flight dynamics]]):
▲* Roll - <math>\phi</math>: rotation about the X-axis
▲* Pitch - <math>\theta</math>: rotation about the Y-axis
▲* Yaw - <math>\psi</math>: rotation about the Z-axis
▲where the X-axis points forward, Y-axis to the right and Z-axis downward and in the example to follow the rotation occurs in the order yaw, pitch, roll (about body-fixed axes).
Nevertheless, it is not easy to find a matrix expression with Tait-Bryan angles because its final expression depends on how the rotations are applied.
== Conversion ==
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