  We present muon-spin rotation measurements on polycrystalline samples of the
complete family of the antiferromagnetic (AF) $zigzag$ chain compounds,
Na$_x$Ca$_{1-x}$V$_2$O$_4$. In this family, we explore the magnetic properties
from the metallic NaV$_2$O$_4$ to the insulating CaV$_2$O$_4$. We find a
critical $x_c(\sim0.833)$ which separates the low and high Na-concentration
dependent transition temperature and its magnetic ground state. In the $x<x_c$
compounds, the magnetic ordered phase is characterized by a single homogenous
phase and the formation of incommensurate spin-density-wave order. Whereas in
the $x>x_c$ compounds, multiple sub-phases appear with temperature and $x$.
Based on the muon data obtained in zero external magnetic field, a careful
dipolar field simulation was able to reproduce the muon behavior and indicates
a modulated helical incommensurate spin structure of the metallic AF phase. The
incommensurate modulation period obtained by the simulation agrees with that
determined by neutron diffraction.
