Note the difference between <math>(v_X,v_Y,v_Z)</math> and <math>(v'_x,v'_y,v'_z)</math>
==Example==
The matrix <math>R</math> describes a rotation around the axis <math>\tfrac 13(\sqrt 3,\sqrt 3,\sqrt 3)</math> through an angle of 90°.
:<math>R = \tfrac 13\begin{pmatrix}
1 & 1  \sqrt 3 & 1 + \sqrt 3 \\
1 + \sqrt 3 & 1 & 1  \sqrt 3 \\
1  \sqrt 3 & 1 + \sqrt 3 & 1
\end{pmatrix}</math>
The vector <math>v=(1,1,0)</math> is actively rotated to the vector
:<math>Rv=R(1,1,0)=\tfrac 13 (\sqrt 3,\sqrt 3,1\sqrt 3)</math>.
A passive rotation by <math>R</math> rotates the coordinate sytem in the de opposie direction, described by the matrix <math>R^{1}=R^*</math>:
:<math>R^* = \tfrac 13\begin{pmatrix}
1 & 1 + \sqrt 3 & 1  \sqrt 3 \\
1  \sqrt 3 & 1 & 1 + \sqrt 3 \\
1 + \sqrt 3 & 1  \sqrt 3 & 1
\end{pmatrix}</math>
The new unit vectors are the columns of this matrix.
The coördinates of <math>v</math> with respect to the new coordinate system are given by:
:<math>(R^*)^{1}v=Rv=R(1,1,0)=\tfrac 13 (\sqrt 3,\sqrt 3,1\sqrt 3)</math>.
Thes are the same numbers as the coördinates of <math>Rv</math> in the active rotation, but with a different meaning.
Under the active rotatie <math>Rv=\tfrac 13 (\sqrt 3,\sqrt 3,1\sqrt 3)</math> means
:<math>Rv=\tfrac 13 \sqrt 3\,(1,0,0)+\tfrac 13 \sqrt 3\,(0,1,0)+\tfrac 13 (1\sqrt 3)\,(0,0,1)</math>,
while under the passive rotation the coördinates have the meaning:
:<math>v=\tfrac 13 \sqrt 3\,(1,1\sqrt 3,1+\sqrt 3)+ \tfrac 13 \sqrt 3\,(1+\sqrt 3,1,1\sqrt 3)+\tfrac 13 (1\sqrt 3)\,(1\sqrt 3,1+\sqrt 3,1)</math>
==See also==
