Euclidean distance (L2)

The straight-line distance between two points, ignoring paths and turns.

The Euclidean distance is the straight-line distance between two points — the direct distance, ignoring paths and turns.

d (A, B) = (x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2}

In 3D space, with A = (x₁, y₁, z₁) and B = (x₂, y₂, z₂):

d (A, B) = (x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2} + (z_{1} - z_{2})^{2}

For the same points $A (3, 5)$ , $B (6, 9)$ : $3^{2} + 4^{2} = 5$ , shorter than the Manhattan distance of 7.

Applications: k-means, k-NN and anomaly detection in ML; image and template matching in computer vision; word-embedding and document similarity in NLP; navigation and motion planning in robotics.

Formula to remember

Expectation

E [X] = x \sum x p (x)

Probability-weighted average.

Quick check · +5 XP

The dot product of (1, 2, 3) and (4, 0, −1) is…

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