We report on transport measurements in the mixed state of YBa2Cu3O7 single crystals using the dc flux transformer geometry. In the vortex fluid regime we have observed the onset of a nonlocal conductivity. At lower temperatures, close to but above the phase transition temperature the vortices become effectively three dimensional in character. This is in sharp contrast to recent flux transformer experiments on Bi2Sr2CaCu2O which have failed to see a nonlocal conductivity, confirming the two-dimensional character at high temperatures of the vortices in that system. These observations suggest that in a finite temperature interval, the vortex state in YBa2Cu3O7 is a line liquid.