## Quick Reference

(in the plane)

Suppose that a Cartesian coordinate system has a given *x*-axis and *y*-axis with origin *O* and given unit length, so that a typical point *P* has coordinates (*x*, *y*). Consider taking a new coordinate system with the same origin *O* and the same unit length, with *X*-axis and *Y*-axis, such that a rotation through an angle α (with the positive direction taken anticlockwise) carries the *x*-axis to the *X*-axis and the *y*-axis to the *Y*-axis. With respect to the new coordinate system, the point *P* has coordinates (*X*, *Y*). Then the old and new coordinates in such a rotation of axes are related by

*x* = *X* cos α − *Y* sin α,

*y* = *X* sin α + *Y* cos α.

In matrix notation, these equations become

and, conversely,For example, in a rotation of axes through an angle of −*π*/4 radians, the coordinates are related byand the curve with equation *x*^{2}−*y*^{2}=2 has equation *XY*=1 in the new coordinate system.

*Subjects:*
Mathematics.

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