Existence of Translation between Each Pair of Points in Euclidean Space

Theorem

Let $\R^n$ denote the real Euclidean space of $n$ dimensions.

Let $\mathbf a = \tuple {a_1, a_2, \ldots, a_n}$ and $\mathbf b = \tuple {b_1, b_2, \ldots, b_n}$ be points in $\R^n$.

There exists an isometry $f: \R^n \to \R^n$ such that $\map f {\mathbf a} = b$.

Proof

Let $\mathbf t = \mathbf a - \mathbf b$.

Then the translation $\tau_\mathbf t$ is such an isometry.

We have that:

$\map {\tau_\mathbf t} {\mathbf a} = \mathbf a - \paren {\mathbf a - \mathbf b} = \mathbf b$

The result follows from Translation Mapping is Isometry.

$\blacksquare$

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 7$: Subspaces and Equivalence of Metric Spaces: Exercise $1$