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=== Passive transformation ===
On the other hand, when one views <math>T</math> as a passive transformation, the initial vector <math>\mathbf{v}=(v_x,v_y,v_z)</math> is left unchanged, while the coordinate system and its basis vectors are transformed in the opposite direction, i.e. with the inverse transformation <math>T^{1}</math>.
<ref name=Amidror>
{{cite book isbn=1402054572 year=2007 publisher=Springer title=The theory of the Moiré phenomenon: Aperiodic layers first=Isaaclast=Amidror▼
url=https://books.google.com/books?id=Z_QRomE5g3QC&pg=PT361 chapter=Appendix D: Remark D.12 page=346 }}▼
</ref> This gives a new coordinate system XYZ with basis vectors:
:<math>\mathbf{e}_X=T^{1}(1,0,0),\ \mathbf{e}_Y=T^{1}(0,1,0),\ \mathbf{e}_Z=T^{1}(0,0,1)</math>
Note the difference between <math>(v_X,v_Y,v_Z)</math> and <math>(v'_x,v'_y,v'_z)</math>
▲{{cite book isbn=1402054572 year=2007 publisher=Springer title=The theory of the Moiré phenomenon: Aperiodic layers first=Isaaclast=Amidror
▲url=https://books.google.com/books?id=Z_QRomE5g3QC&pg=PT361 chapter=Appendix D: Remark D.12 page=346 }}
==See also==

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