Rotation of Cartesian Axes around Vector

Theorem

Let $\mathbf r$ be a vector in space.

Let a Cartesian plane $\CC$ be established such that:

the initial point of $\mathbf r$ is at the origin $O$

the terminal point of $\mathbf r$ is the point $P$.

Let $\tuple {X, Y}$ be the coordinates of $P$ under $\CC$.

Let $\CC$ be rotated about $O$ to $\CC'$, through an angle $\varphi$ in the anticlockwise direction, while keeping $\mathbf r$ fixed.

Let $\tuple {X', Y'}$ be the coordinates of $P$ under $\CC'$.

Then:

$\ds X'$

$=$

$\ds X \cos \varphi + Y \sin \varphi$

$\ds Y'$

$=$

$\ds -X \sin \varphi + Y \cos \varphi$

With reference to the above diagram:

$X P X' = \varphi$

and so:

$OX' = OX \cos \varphi + PX \sin \varphi$

and:

$OY' = OY \cos \varphi - PY \cos \varphi$

Hence the result.

$\blacksquare$

1970: George Arfken: Mathematical Methods for Physicists (2nd ed.) ... (previous) ... (next): Chapter $1$ Vector Analysis $1.2$ Rotation of Coordinates