Active and passive transformation: Difference between revisions

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=== Passive transformation ===

On the other hand, when one views ''<math>R''</math> as a passive transformation, the initial vector '''v''' is left unchanged, while the coordinate system and its basis vectors are rotated in the opposite directon, i.e. with the rotation <math>R^{-1}=R^*</math>. In order for the vector to remain fixed, the coordinates in terms of the new basis must change., Foraccording ato: counterclockwise rotation of coordinate systems:

:<math>\mathbf{v}=(v_1,v_2)=v'_1R^*(1,0)+v'_2R*^(0,1)=R^*(v'_1,v'_2)</math>.

From this equation one sees that the new coordinates (that is, coordinates with respect to the new basis) are given by

:<math>(v'_1,v'_2)=R^{-1}(v_1,v_2)</math>.

so that the coordinates of the vector ''must'' transform according to the inverse of the active transformation operatorof the coordinate system.<ref name=Amidror>

{{cite book |isbn=1-4020-5457-2 |year=2007 |publisher=Springer |title=The theory of the Moiré phenomenon: Aperiodic layers |first=Isaac|last=Amidror