quality:correction, information:filtering
Distance and length

The distance (more precisely the Euclidean distance) between two points of a Euclidean space is the norm of the translation vector that maps one point to the other; that is

d (P, Q) = ‖ P Q → ‖. ({\displaystyle d(P,Q)={\Bigl \|}{\overrightarrow {PQ}}{\Bigr \|}.{\vphantom {\frac {(}{}}}}

The length of a segment PQ is the distance d (P, Q) between its endpoints P and Q. It is often denoted | P Q | {\displaystyle |PQ|}.

The distance is a metric, as it is positive definite, symmetric, and satisfies the triangle inequality

d (P, Q) ≤ d (P, R) + d (R, Q). {\displaystyle d(P,Q)\leq d(P,R)+d(R,Q).}

Moreover, the equality is true if and only if a point R belongs to the segment PQ. This inequality means that the length of any edge of a triangle is smaller than the sum of the lengths of the other edges. This is the origin of the term triangle inequality.

With the Euclidean distance, every Euclidean space is a complete metric space.