Euclidean Distance
Straight-line distance between two points, the L2 norm of the vector between them.
Think of It Like This
What a bird flies, not what a taxi drives.
Euclidean distance is — Pythagoras, extended to as many dimensions as you have. It is the default notion of "close" in k-nearest neighbours, k-means, and most clustering.
The property that matters in practice is that it is sensitive to scale. A feature measured in metres and one measured in millimetres contribute wildly unequally, so the largest-range feature dominates the distance and effectively decides your neighbours. Standardising features before any distance-based method is not optional.
It also degrades in very high dimensions: distances between random points concentrate into a narrow band, so nearest and farthest neighbours become hard to distinguish. That concentration is one reason cosine similarity is preferred for embeddings — though for unit-length vectors the two produce the same ranking, since .