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In \mathbb{R}^3 ( Euclidean Space ), a rotation around the ''x''-axis is given by:
:
\mathcal{R}(\gamma):=
\begin{pmatrix}
1 & 0 & 0 \
0 & \cos{\gamma} & - \sin{\gamma} \
0 & \sin{\gamma} & \cos{\gamma}
\end{pmatrix}
(where \gamma is the Roll Angle )

A rotation around the ''y''-axis is given by:
:
\mathcal{P}(\beta):=
\begin{pmatrix}
\cos{\beta} & 0 & \sin{\beta} \
0 & 1 & 0 \
- \sin{\beta} & 0 & \cos{\beta}
\end{pmatrix}
(where \beta is the Pitch Angle )

A rotation about the ''z''-axis is given by:
:
\mathcal{Y}(\alpha):=
\begin{pmatrix}
\cos{\alpha} & - \sin{\alpha} & 0 \
\sin{\alpha} & \cos{\alpha} & 0 \
0 & 0 & 1
\end{pmatrix}
(where \alpha is the Yaw Angle )

with the equivalent counter-clockwise rotation in \mathbb{R}^2 can be given from the Minor \mathcal{Y}( heta)_{3,3} of the rotation around the ''z''-axis:
:
\begin{pmatrix}
\cos{ heta} & -\sin{ heta} \
\sin{ heta} & \cos{ heta}
\end{pmatrix}


Notice that this can be viewed as rotating the axes clockwise or the vector counterclockwise.

Using the Yaw-pitch-roll System any 3-dimensional rotation matrix \mathcal{M}\in\mathbb{R}^{3 imes 3} can be characterised by the three angles \alpha, \beta and \gamma:
:\mathcal{M} is rotation matrix in \mathbb{R}^{3 imes 3}\,\Leftrightarrow\,\exist\,\alpha,\beta,\gamma\in[0\ldots\pi):\,\mathcal{M}=\mathcal{Y}(\alpha)\,\mathcal{P}(\beta)\,\mathcal{R}(\gamma)

See also the General Formula For A 3 × 3 Rotation Matrix In Terms Of The Axis And The Angle .

The set of all rotations about a given axis, together with the operation of Composition , form a Continuous Group . The matrices discussed in this article then provide a Representation of the group.

\mathcal{M}\in\mathbb{R}^{3 imes 3} is a rotation matrix if and only if \mathcal{M} is Orthonormal .

\mathcal{M} is Orthonormal , if its column vectors are forming an Orthonormal Basis of \mathbb{R}^{3 imes 3}, that is, the scalar product between any two column vectors is ''0'' ( Orthogonality ) and the scalar product of a column vector with itself is ''1'' ( Normalized Vector ). In other words the inverse of a rotation matrix is its Transpose :
:\mathcal{M}\,\mathcal{M}^{-1}=\mathcal{M}\,\mathcal{M}^ op=\mathcal{I} where \mathcal{I} is the identity matrix.

In particular, the Dot Product of each row with itself is equal to 1, and the dot product of different rows is zero.


SEE ALSO



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