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(Inserted {{short descriptionDistinction between meanings of Euclidian space transformations}}) 
m (Typo/general fixes, replaced: Euclidian → Euclidean (3)) 

{{short descriptionDistinction between meanings of
{{Forthe concept of "passive transformation" in grammaractive voicepassive voice}}
[[File:PassiveActive.JPGthumb310pxIn the active transformation (left), a point moves from position P to P' by rotating clockwise by an angle θ about the origin of the coordinate system. In the passive transformation (right), point P does not move, while the coordinate system rotates counterclockwise by an angle θ about its origin. The coordinates of P' in the active case (that is, relative to the original coordinate system) are the same as the coordinates of P relative to the rotated coordinate system.]]
In [[analytic geometry]], spatial transformations in the 3dimensional
Put differently, a ''passive'' transformation refers to description of the ''same'' object in two different coordinate systems.<ref name= Davidson>
which can be viewed either as an ''active transformation'' or a ''passive transformation'' (where the above matrix will be inverted), as described below.
==Spatial transformations in the
In general a spatial transformation <math>T\colon\R^3\to \R^3</math> may consist of a translation and a linear transformation. In the following, the translation will be omitted, and the linear transformation will be represented by a 3×3matrix <math>T</math>.
===Active transformation===
As an active transformation, <math>T</math> transforms the initial vector <math>\mathbf{v}=(v_x,v_y,v_z)</math> into a new vector <math>\mathbf{v}'=(v'_x,v'_y,v'_z)=T\mathbf{v}=T(v_x,v_y,v_z)</math>.
If one views <math>\{\mathbf{e}'_x=T(1,0,0),\ \mathbf{e}'_y=T(0,1,0),\ \mathbf{e}'_z=T(0,0,1)\}</math> as a new basis, then the coordinates of the new vector <math>\mathbf{v}'=v_x\mathbf{e}'_x+v_y\mathbf{e}'_y+v_z\mathbf{e}'_z</math> in the new basis are the same as those of <math>\mathbf{v}=v_x\mathbf{e}_x+v_y\mathbf{e}_y+v_z\mathbf{e}_z</math> in the original basis. Note that active transformations make sense even as a linear transformation into a different vector space. It makes sense to write the new vector in the unprimed basis (as above) only when the transformation is from the space into itself.
