K3 surfaces with Picard number three and canonical vector heights

In this paper we construct the first known explicit family of K3 surfaces defined over the rationals that are proved to have geometric Picard number $3$. This family is dense in one of the components of the moduli space of all polarized K3 surfaces with Picard number at least $3$. We also use an example from this family to fill a gap in an earlier paper by the first author. In that paper, an argument for the nonexistence of canonical vector heights on K3 surfaces of Picard number $3$ was given, based on an explicit surface that was not proved to have Picard number $3$. We redo the computations for one of our surfaces and come to the same conclusion.