Composing geometric transformations

Solving a geometry problem, whether in the plane or in three-dimensional space, can sometimes prove tricky. It often helps to turn it into a vector problem and bring in suitably chosen transformations. These can then be composed… or decomposed, allowing vector algebra to make the conclusion easier to reach.

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Transformations of points and vectors -------------------------------------

In the plane or in space, a transformation is first and foremost a bijection: it assigns each point of the plane (or of space) a unique point of the plane (or of space). Some geometric transformations are already familiar to us. Less familiar, however, are the associated vector transformations: we move from a space of points – which mathematicians call an affine space – to a vector space. Each transformation of a point space (in the plane or in space) gives rise to a vector transformation: if f is a transformation that sends every point A to A' and every point B to B', then the associated vector transformation $\phi \backslash varphi$ is defined by $\phi (\overrightarrow{AB})=\overrightarrow{A^{\mathrm{\prime }}{B}^{\mathrm{\prime }}}\backslash varphi\; (\backslash overrightarrow\{AB\})\; =\; \backslash overrightarrow\{A\text{'}B\text{'}\}$ and is easily shown to be linear, meaning that $\phi (a\overrightarrow{u}+b\overrightarrow{v})=a\phi (\overrightarrow{u})+b\phi (\overrightarrow{v})\backslash varphi\; (a\; \backslash overrightarrow\{u\}\; +\; b\; \backslash overrightarrow\{v\})\; =\; a\; \backslash varphi\; (\backslash overrightarrow\{u\}\; )\; +\; b\; \backslash varphi\; (\backslash overrightarrow\{v\}\; )$ for every pair (a, b) of real numbers.