> For the complete documentation index, see [llms.txt](https://wiki.ariitk.in/wiki/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://wiki.ariitk.in/wiki/concepts/rotations.md).

# Rotations

Rotating frames.

A vector whose x,y,z components are $$x,y,z$$ respectively is represented as the column matrix$$\begin{bmatrix}x\y\z\end{bmatrix}$$ 

### Rotation of vectors in 2D:

In two dimensions, the standard rotation matrix has the following form:-

​<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0166e674df67cf24314537211848adec91813945" alt="alt text" data-size="original">​

The above matrix represents the counterclockwise rotation of a vector through angle θ. The vector in this figure is initially aligned with the x-axis

![alt text](https://upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Counterclockwise_rotation.png/220px-Counterclockwise_rotation.png)​

If a vector $$\begin{bmatrix}x\y\end{bmatrix}$$ is rotated counterclockwise by $$\theta$$, the resultant vector $$\begin{bmatrix}x'\y'\end{bmatrix}$$ is given as

​![alt text](https://wikimedia.org/api/rest_v1/media/math/render/svg/657b520ec337f95a996bc9e77f07401778d272af) ![alt text](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddafa97cf937c752708b51b3ba65d9e4e797e6c5)​

​![alt text](https://wikimedia.org/api/rest_v1/media/math/render/svg/50622f9a4a7ba2961f5df5f7e0882983cf2f1d2f)​

### Rotation of vectors in 3D:

The following matrices represent counterclockwise rotations by angle $$\theta$$ about the x,y and z axes respectively.

​![alt text](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6821937d5031de282a190f75312353c970aa2df)​

Their usage is similiar as that explained in the previous section.

For the applications of rotations in UAVs, refer to the following link:

{% file src="/files/-LVOtaac9JqoLnqMiRsi" %}
Flight Mechanics
{% endfile %}


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