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Rotation in Two Dimensions

Suppose we have a point A in a plane with perpendicular  x  and  y  coordinate axes scribed on it, as pictured in Fig. 23.4.

  

Figure: A fixed point A can be located in a plane using either of two coordinate systems O (x,y) and O' (x',y') that differ from each other by a rotation of  $\theta$  about the common origin (0,0).

We can scribe a different pair of perpendicular coordinate axes  x'  and  y'  on the same plane surface using dashed lines by simply rotating the original coordinate axes by an angle  $\theta$  about their common origin, the coordinates of which are (0,0) in either coordinate system.

Now suppose that we have the coordinates  (x_A,y_A)  of point A in the original coordinate system and we would like to transform these coordinates into the coordinates   (x'_A,y'_A)  of the same point in the new coordinate system.^23.14 How do we do it? By trigonometry, of course. You can figure this out for yourself. The transformation is

$$x \, \cos(\theta) \; + \; y \, \sin(\theta)$$ x' = (23.2)

$$-x \, \sin(\theta) \; + \; y \, \cos(\theta)$$ y' = (23.3)