Euclidean Distance¶

Definition¶

The Euclidean distance between two points $\boldsymbol{x}=(x_1,\ldots,x_n)$ and $\boldsymbol{y}=(y_1,\ldots,y_n)$ in $\mathbb{R}^n$ is $d(\boldsymbol{x},\boldsymbol{y})=\|\boldsymbol{x}-\boldsymbol{y}\|=\sqrt{(x_1-y_1)^2+\cdots+(x_n-y_n)^2}$$\sqrt{(x_1-y_1{)}^2+\cdots +(x_n-y_n{)}^2}$.

Interpretation¶

Euclidean distance is the ordinary straight-line distance, inherited from the Euclidean norm: the distance between two points is the length of the vector from one to the other.

From $\boldsymbol{P}=(1,1)$ to $\boldsymbol{Q}=(4,3)$ the coordinates change by $3$ horizontally and $2$ vertically. These differences are the legs of a right triangle, so by Pythagoras $d(\boldsymbol{P},\boldsymbol{Q})=\sqrt{3^2+2^2}=\sqrt{13}$$\sqrt{13}$. In $\mathbb{R}^n$ the formula adds one squared difference per coordinate.

The squares remove the signs, so $d(\boldsymbol{x},\boldsymbol{y})=d(\boldsymbol{y},\boldsymbol{x})$$d(\mathbf{y},\mathbf{x})$, and $d(\boldsymbol{x},\boldsymbol{y})=0$ only when $\boldsymbol{x}=\boldsymbol{y}$$\mathbf{y}$. For $n=1$ the formula reduces to $|x_1-y_1|$$y_1|$, the distance on the real line given by the absolute value. A detour is never shorter: $d(\boldsymbol{x},\boldsymbol{z})\le d(\boldsymbol{x},\boldsymbol{y})+d(\boldsymbol{y},\boldsymbol{z})$ is the triangle inequality for the norm. OR Modelling uses these distances as edge lengths in a Euclidean network.

Key Results¶

• The distance between two points is the length of the vector connecting them.
• The distance from a point to a line is the minimum Euclidean distance from the point to points on the line.

Examples¶

• $d((0,0),(3,4))=5$$5$.
• $d((1,1),(2,2))=\sqrt{2}$$\sqrt{2}$.

Prerequisites¶

• Euclidean Norm

Related Concepts¶

• Euclidean Space
• Inner Product

Sources¶

• Math II: Linear Algebra · Week 5

Aliases: Euclidean metric, L2 distance Keywords: straight-line distance Symbols: real numbers, square root

Last updated: 20 October 2026 03:02