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Mathematics-Online lexicon:

Rotation of a Cartesian Coordinate System

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Rotating the $ xy$ -plane by the angle $ \alpha$ about the $ z$ -axis the coordinates of a point $ P=(p_1,p_2,p_3)$ transform as follows:

$\displaystyle x' = \cos(\alpha)\,x\, + \,\sin(\alpha)\,y,\quad
y' = -\sin(\alpha)\,x\, + \,\cos(\alpha)\,y,\quad
z' = z\,


Analogous formulas are obtained by rotations about the $ yz$ - and the $ zx$ -plane.


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  automatically generated 3/28/2008