  Many classical objects on a surface S can be interpreted as cross-ratio
functions on the circle at infinity of the universal covering. This includes
closed curves considered up to homotopy, metrics of negative curvature
considered up to isotopy and, in the case of interest here, tangent vectors to
the Teichm\"uller space of complex structures on S. When two cross-ratio
functions are sufficiently regular, they have a geometric intersection number,
which generalizes the intersection number of two closed curves. In the case of
the cross-ratio functions associated to tangent vectors to the Teichm\"uller
space, we show that two such cross-ratio functions have a well-defined
geometric intersection number, and that this intersection number is equal to
the Weil-Petersson scalar product of the corresponding vectors.
