Euclidean metric

Example

Let $n \ge 1$ be an integer. For points $x = (x_1, \ldots, x_n)$ and $y = (y_1, \ldots, y_n)$ in $\mathbb{R}^n$, let $$d_2(x, y) = \sqrt{\sum_{i=1}^{n} (x_i - y_i)^2}.$$ Then $d_2$ is a metric on $\mathbb{R}^n$, the Euclidean metric.

Remarks

For $n = 1$, $d_2(x, y) = |x - y|$, the usual metric on the real line. For $n = 2$, the formula is the theorem of Pythagoras for the segment from $x$ to $y$. The triangle inequality for $d_2$ follows from the Cauchy–Schwarz inequality.

Explanations

L1 human

The Euclidean metric is the length of the straight segment between two points. In the plane, the distance from $(0, 0)$ to $(3, 4)$ is $\sqrt{9 + 16} = 5$.