  We introduce the relation ${\rho}_{\lambda}$-orthogonality in the setting of
normed spaces as an extension of some orthogonality relations based on norm
derivatives, and present some of its essential properties. Among other things,
we give a characterization of inner product spaces via the functional
${\rho}_{\lambda}$. Moreover, we consider a class of linear mappings preserving
this new kind of orthogonality. In particular, we show that a linear mapping
preserving ${\rho}_{\lambda}$-orthogonality has to be a similarity, that is, a
scalar multiple of an isometry.
