Understanding the mathematical properties of document vectors is crucial for text mining. When you represent documents as vectors — such as with bag-of-words or TF-IDF — these vectors live in a high-dimensional space, where each dimension corresponds to a unique term. To analyze and compare these vectors, you need to understand vector norms, angles, and the concept of sparsity.

A vector norm is a measure of a vector's length or magnitude. In document spaces, two norms are most common: L1 norm and L2 norm. The L1 norm (also called "Manhattan" or "taxicab" norm) sums the absolute values of all entries in the vector. For a document vector, this means adding up the absolute frequencies or weights of all terms present in the document. The L2 norm (or "Euclidean" norm) is the square root of the sum of the squares of all entries. This norm reflects the straight-line distance from the origin to the point representing the document in the vector space. In practice, the L1 norm can be interpreted as the total "amount" of words (or weight) in a document, while the L2 norm gives a sense of the document's overall "magnitude" in the space, which is useful when comparing documents of different lengths or when normalizing vectors for similarity calculations.

The angle between vectors is another important property when comparing documents. In high-dimensional spaces, the angle between two document vectors reflects how similar their directions are, regardless of their length. A small angle means the documents share similar patterns of term usage, while a large angle indicates they are quite different. The cosine of the angle is often used as a measure of similarity, since it remains unaffected by the overall length of the documents. When two document vectors point in the same direction, their angle is zero and the cosine similarity is one, indicating maximum similarity. If they are orthogonal (at 90 degrees), the cosine similarity is zero, meaning they share no terms.

Sparsity is a defining characteristic of document-term matrices. In most practical cases, each document contains only a tiny fraction of all possible terms in the vocabulary. This means that most entries in a document vector are zero. High-dimensional sparsity has several implications: geometrically, most document vectors are nearly orthogonal to each other, because the chance of two documents sharing many rare terms is low. Computationally, sparsity is beneficial — it allows you to store and process large document collections efficiently by focusing only on the nonzero entries. However, sparsity can also make it harder to find meaningful similarities, especially when documents are short or the vocabulary is very large. Understanding these properties helps you design and interpret algorithms for document comparison and retrieval in text mining applications.