What is a vector norm?

A vector norm is a single number describing a vector’s overall size or length, calculated from its individual dimension values, most commonly using the same formula that produces Euclidean distance when applied to the difference between two vectors instead of to a single vector on its own.

How is a vector’s norm actually calculated?

The most common norm, generally called the L2 norm or Euclidean norm, squares each of a vector’s individual values, adds these squares together, and takes the square root of the total — exactly the same calculation used for Euclidean distance, just applied to a single vector’s own values rather than to the difference between two separate vectors. Other norms exist too, following the same pattern as the distance functions covered elsewhere in this glossary: an L1 norm sums the absolute values of a vector’s dimensions instead of squaring them, mirroring Manhattan distance, and the general family these belong to follows the same Minkowski pattern described in this glossary’s page on Minkowski distance.

Why do different norms correspond to the distance functions covered elsewhere in this glossary?

A norm and a distance function are closely related ideas: applying a chosen norm to the difference between two vectors is exactly how the corresponding distance function is defined. The L2 norm applied to a difference produces Euclidean distance; the L1 norm applied to a difference produces Manhattan distance. This isn’t a coincidence — a norm measures how large a single vector is, and a distance function measures how large the gap between two vectors is, which is naturally computed by first finding that gap as a new vector and then measuring its size using a norm.

Why does a vector’s norm matter for normalization specifically?

Normalizing a vector means dividing every one of its individual values by its own norm, producing a new vector that points in exactly the same direction as the original but has a length of exactly 1. This step is what makes cosine similarity and normalized inner product produce identical rankings, as covered in this site’s page on cosine similarity — once every vector in a comparison has been normalized this way, the norm itself becomes a fixed constant that no longer affects the outcome, leaving only direction to distinguish one vector from another.

Vector norm is the mathematical foundation underlying both Euclidean distance and vector normalization, both covered elsewhere in this glossary. From here, the pages on vector normalization and on cosine similarity are the most useful next steps for seeing how a vector’s norm actually gets used in practice.