Rotation of Cartesian Axes around Vector
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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:
\(\ds X'\) | \(=\) | \(\ds X \cos \varphi + Y \sin \varphi\) | ||||||||||||
\(\ds Y'\) | \(=\) | \(\ds -X \sin \varphi + Y \cos \varphi\) |
Proof
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.
$\blacksquare$
Sources
- 1970: George Arfken: Mathematical Methods for Physicists (2nd ed.) ... (previous) ... (next): Chapter $1$ Vector Analysis $1.2$ Rotation of Coordinates