Mapping/Examples/Rotation through 30 Degrees
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Example of Mapping
Let $\Gamma$ be the Cartesian plane.
The rotation $R_{30 \degrees}$ of $\Gamma$ anticlockwise through an angle of $30 \degrees$ about the origin $O$ is a mapping from $\Gamma$ to $\Gamma$.
Proof
For every point $P$ in $\Gamma$ which is not the origin $O$ it is possible to:
- construct a straight line $OP$
- construct a straight line $OP'$ at an angle of $30 \degrees$ to $OP$ measured in an anticlockwise direction
- construct the point $P'$ such that $\len \paren {OP} = \len \paren {OP'}$.
It can be seen that for every $P$ there is a unique $P'$ to which $R_{30 \degrees}$ maps $P$, apart from $O$ which is mapped to $O$.
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 20 \ (2)$: Introduction