Rotation of Axes

Rotation of Axes

Rotation of axes keeps the origin fixed but turns the pair of perpendicular axes through a fixed angle $\theta$, measured counter-clockwise from the old axes to the new axes. A point $P$ in the plane does not move, but its coordinates change because the reference directions have turned.

Let $P$ have old coordinates $(x,y)$ and new coordinates $(x',y')$ after the axes are rotated through angle $\theta$. Resolving the position of $P$ along the new $x'$ and $y'$ directions (each of which makes angle $\theta$ with the corresponding old axis) gives the rotation formulas expressing the old coordinates in terms of the new ones:

These may be written compactly using a rotation matrix:

Because a rotation matrix is orthogonal, its inverse equals its transpose, so solving for the new coordinates in terms of the old ones simply changes the sign pattern:

$x' = x\cos\theta + y\sin\theta, \qquad y' = -x\sin\theta + y\cos\theta$ …