Euclidean distance measures the straight-line separation between two points or vectors. In an orthogonal, rectilinear vector space, subtract corresponding coordinates and take the square root of their squared differences: the Pythagorean theorem provides the calculation. section 3.2.3 section 3.3
Comparing two term-frequency vectors
Represent two documents as Vec1 = (1, 0) and Vec2 = (2, 1), where coordinates can represent word frequencies. Subtract one vector from the other, then calculate the length of the resulting difference vector. That length is their Euclidean distance. section 3.3
Euclidean distance is also called Cartesian distance, the L2 distance, or the 2-norm. It works naturally for a small number of dimensions, but high-dimensional NLP vectors often use cosine distance instead because Euclidean magnitudes can become unwieldy. section 3.3 section 4.6
Coordinate geometry matters: latitude and longitude lie on an approximately spherical surface, so ordinary Euclidean distance is unsuitable for long geographic routes, although it can approximate nearby locations. section 3.2.3
