Applications Vector space model

relevance rankings of documents in keyword search can calculated, using assumptions of document similarities theory, comparing deviation of angles between each document vector , original query vector query represented same kind of vector documents.

in practice, easier calculate cosine of angle between vectors, instead of angle itself:

cos
⁡

θ

=





d

2



⋅

q




∥


d

2



∥


∥

q

∥






{\displaystyle \cos {\theta }={\frac {\mathbf {d_{2}} \cdot \mathbf {q} }{\left\|\mathbf {d_{2}} \right\|\left\|\mathbf {q} \right\|}}}

where





d

2



⋅

q



{\displaystyle \mathbf {d_{2}} \cdot \mathbf {q} }

intersection (i.e. dot product) of document (d2 in figure right) , query (q in figure) vectors,




∥


d

2



∥



{\displaystyle \left\|\mathbf {d_{2}} \right\|}

norm of vector d2, ,




∥

q

∥



{\displaystyle \left\|\mathbf {q} \right\|}

norm of vector q. norm of vector calculated such:

∥

q

∥

=



∑

i
=
1


n



q

i


2






{\displaystyle \left\|\mathbf {q} \right\|={\sqrt {\sum _{i=1}^{n}q_{i}^{2}}}}

as vectors under consideration model elementwise nonnegative, cosine value of 0 means query , document vector orthogonal , have no match (i.e. query term not exist in document being considered). see cosine similarity further information.