Cosine Coefficient

The cosine coefficient is used to obtain a measure of similarity between query vector and the various documents in a collection. The same measure could be used to the determination of two documents. Thus, given the term vectors for two documents, DOC_i and DOC_j , the similarity between them may be defined as:

COSINE(DOC_i , DOC_j) = A/B
where
A = Sum(Term _ik . Term _jk)
B = Sqt.(Sum(Term _ik)²) x Sqt.(Sum(Term _jk)²)
(where k = 1,...,t)

The concept is from Vector inner products (dot products), Let say two vectors R and Q are defined as follows:
R = R₁ + R₂ + ... + R_t
Q = Q₁ + Q₂ + ... + Q_t

The dot product is:

1) The Component method:
R.Q = R₁ Q₁ + R₂Q₂ + ... + R_tQ_t

2) The Magnitude/Angle method:
R.Q = |R| . |Q| . cos(θ)

Thus:
|R| . |Q| . cos(θ) = R₁ Q₁ + R₂Q₂ + ... + R_tQ_t
cos(θ) = (R₁ Q₁ + R₂Q₂ + ... + R_tQ_t) / |R| . |Q|