Euclidean distance

Definition

Let $n$ be a positive integer, and let $p = (p_1, \dots, p_n)$ and $q = (q_1, \dots, q_n)$ be points of $\mathbb{R}^n$. The Euclidean distance between $p$ and $q$ is the real number $$d(p, q) = \|p - q\| = \sqrt{(p_1 - q_1)^2 + (p_2 - q_2)^2 + \dots + (p_n - q_n)^2},$$ where $\|p - q\|$ is the length of the vector $p - q$.

Remarks

The function $d$ is a metric on $\mathbb{R}^n$. For all points $p$, $q$ and $r$: $d(p, q) \geq 0$, $d(p, q) = 0$ if and only if $p = q$, $d(p, q) = d(q, p)$, and $d(p, r) \leq d(p, q) + d(q, r)$. For $n = 1$ the formula is $d(p, q) = |p - q|$. Some applications use the squared distance $d(p, q)^2$, because it has no square root. The squared distance is not a metric, because it does not satisfy the triangle inequality.

Explanations

L1 human

To find the distance between two points in the plane, draw the right triangle with horizontal and vertical legs between them. The legs have lengths $|p_1 - q_1|$ and $|p_2 - q_2|$. The distance is the hypotenuse, so the Pythagorean theorem gives the formula. For $(1, 1)$ and $(4, 5)$ the distance is $\sqrt{9 + 16} = 5$.