Euclidean Metric on Real Number Plane is Rotation Invariant

Theorem
Let $r_\alpha: \R^2 \to \R^2$ denote the rotation of the Euclidean plane about the origin through an angle of $\alpha$.

Let $d_2$ denote the Euclidean metric on $\R^2$.

Then $d_2$ is unchanged by application of $r_\alpha$:


 * $\forall x, y \in \R^2: \map {d_2} {\map {r_\alpha} x, \map {r_\alpha} y} = \map {d_2} {x, y}$

Proof
Let $x = \tuple {x_1, x_2}$ and $y = \tuple {y_1, y_2}$ be arbitrary points in $\R^2$.

Note that $\paren {\map {d_2} {x, y} }^2$ can be expressed as:
 * $\paren {\map {d_2} {x, y} }^2 = \paren {\mathbf x - \mathbf y}^\intercal \paren {\mathbf x - \mathbf y}$

where:
 * $x$ and $y$ are expressed in vector form: $\mathbf x = \begin {pmatrix} x_1 \\ x_2 \end {pmatrix}, y = \begin {pmatrix} y_1 \\ y_2 \end {pmatrix}$
 * $\paren {\mathbf x - \mathbf y}^\intercal$ denotes the transpose of $\paren {\mathbf x - \mathbf y}$

Then: