Editing Rotation matrix (section)

===Basic 3D rotations===

[[File:Angle-orientation-in-rotation-matrix-around-x-y-and-z.svg|thumb]]
A basic 3D rotation (also called elemental rotation) is a rotation about one of the axes of a coordinate system. The following three basic rotation matrices rotate vectors by an angle {{mvar|θ}} about the {{mvar|x}}-, {{mvar|y}}-, or {{mvar|z}}-axis, in three dimensions, using the [[right-hand rule]]—which codifies their alternating signs.<ref>{{cite web | url=https://alphons.io/question/5497/what-are-the-rotation-matrix-in-3d | title=What are the rotation matrices in 3D ? }}</ref> Notice that the right-hand rule only works when multiplying <math>R \cdot        \vec{x}</math>. (The same matrices can also represent a clockwise rotation of the axes.<ref group=nb>Note that if instead of rotating vectors, it is the reference frame that is being rotated, the signs on the {{math|sin ''θ''}} terms will be reversed. If reference frame A is rotated anti-clockwise about the origin through an angle {{mvar|θ}} to create reference frame B, then {{mvar|R<sub>x</sub>}} (with the signs flipped) will transform a vector described in reference frame A coordinates to reference frame B coordinates. Coordinate frame transformations in aerospace, robotics, and other fields are often performed using this interpretation of the rotation matrix.</ref>)



:<math>
\begin{alignat}{1}
R_x(\theta) &= \begin{bmatrix}
1 &  0            &  0           \\
0 &  \cos \theta  & -\sin \theta \\[3pt]
0 &  \sin \theta & \cos \theta \\[3pt]
\end{bmatrix} \\[6pt]
R_y(\theta) &= \begin{bmatrix}
\cos \theta & 0 & \sin \theta \\[3pt]
0           & 1 &  0           \\[3pt]
-\sin \theta & 0 &  \cos \theta \\
\end{bmatrix} \\[6pt]
R_z(\theta) &= \begin{bmatrix}
\cos \theta & -\sin \theta & 0 \\[3pt]
\sin \theta &  \cos \theta & 0 \\[3pt]
0           &  0           & 1 \\
\end{bmatrix}
\end{alignat}
</math>

For [[column vector]]s, each of these basic vector rotations appears counterclockwise when the axis about which they occur points toward the observer, the coordinate system is right-handed, and the angle {{mvar|θ}} is positive. {{math|''R''<sub>''z''</sub>}}, for instance, would rotate toward the {{nowrap|{{math|''y''}}-axis}} a vector aligned with the {{nowrap|{{math|''x''}}-axis}}, as can easily be checked by operating with {{math|''R''<sub>''z''</sub>}} on the vector {{math|(1,0,0)}}:
:<math> R_z(90^\circ) \begin{bmatrix} 1 \\ 0 \\ 0 \\ \end{bmatrix} =
\begin{bmatrix} \cos 90^\circ &  -\sin 90^\circ & 0 \\ \sin 90^\circ &  \quad\cos 90^\circ & 0\\ 0 & 0 & 1\\ \end{bmatrix} 
\begin{bmatrix} 1 \\ 0 \\ 0 \\ \end{bmatrix} =
\begin{bmatrix} 0 &  -1 & 0 \\ 1 & 0 & 0\\ 0 & 0 & 1\\ \end{bmatrix} 
\begin{bmatrix} 1 \\ 0 \\ 0 \\ \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \\ 0 \\ \end{bmatrix}
</math>

This is similar to the rotation produced by the above-mentioned two-dimensional rotation matrix. See [[#Ambiguities|below]] for alternative conventions which may apparently or actually invert the sense of the rotation produced by these matrices.