Orthogonal matrices and euclidean distances

Question

Recall that the Euclidean distance between two points x, y ∈ R^3 is |x − y|, where |z|^2 = z^T*z, for any z ∈ R^3 , thought of as a column vector.

(i) If A ∈ M3(R) is orthogonal, show that the map φA : R^3 → R^3 : x → Ax preserves Euclidean distance, in the sense that |Ax − Ay| = |x − y|, for all x, y ∈ R^3.

(ii) If v ∈ R^3 show that the translation map τv : R^3 → R^3 : x → x + v preserves Euclidean distance.

(iii) Deduce that a general Euclidean transformation ψA,v : R^3 → R^3 : x → Ax + v, for some orthogonal A ∈ M3(R) and v ∈ R^3 , preserves Euclidean distance.

I have already proved the first part using the fact that |a|^2=a^T*a and the fact that A is orthogonal. But I do not know how to prove part (ii) and I assume that part (iii) follows from the previous parts

iv) Show that the set of all Euclidean transformations is closed under composition, that the identity map IdR^3 is a Euclidean transformation and that the inverse of a Euclidean transformation is also a Euclidean transformation.

Answer 1

For (2), you can just calculate directly the distance between $x'=x+v$ and $y'=y+v$ $$|x'-y'|=|x+v-(y+v)|=|x-y|$$

For (3), simply use the properties of (1) and (2): $$|x'-y'|=|Ax+v-(Ay+v)|=|Ax-Ay|=|A(x-y)|=|(x-y)|$$

Answer 2

For the second part, you just use that fact that $\bigl\lVert(x+v)-(y+v)\bigr\rVert=\lVert x-y\rVert$. For the third one, the map $x\mapsto Ax+v$ is the composition of the masps from the first two parts, each of which preserves the distance. So, their composition preserves the distance too.