In the figure below, rotation of axis system X0, Y0 about the z axis (which is perpendicular to the plane of X0, Y0 by an angle θ is accomplished by the rotation matrix:

1
\(\begin{pmatrix} \cos \theta & \sin \theta & 0 \\\ -\sin \theta & \cos \theta & 0 \\\ 0 & 0 & 1\end{pmatrix}\)
2
\(\begin{pmatrix} \cos \theta & -\sin \theta & 1 \\\ \sin \theta & \cos \theta & 1 \\\ 1 & 1 & 0\end{pmatrix}\)
3
\(\begin{pmatrix} \cos \theta & -\sin \theta & 0 \\\ \sin \theta & \cos \theta & 0 \\\ 0 & 0 & 1\end{pmatrix}\)
4
\(\begin{pmatrix} \cos \theta & \sin \theta & 1 \\\ -\sin \theta & \cos \theta & 1 \\\ 1 & 1 & 0\end{pmatrix}\)

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