Rotations using quaternions

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Usually rotations are defined by 3 angles, either Euler or Roll-Pitch-Yaw angles. So three successive rotations around three different axes lead to a combined rotation, that can be described by a single rotation matrix. Such a combined rotation is equal to a rotation around a certain axis in three-dimensional space about a certain angle. This is shown in the figure on the right. The vector $\vec{\mathbf{r}}$ is rotated such that it results in $\vec{\mathbf{r}}'$. This could be done by rotating the vector around the z-, the y- and the x-axis successively or by just rotating it around the rotation axis $\vec{\mathbf{u}}$ by $\phi$. The result is the same as you can see in the example below.