In particle physics, the vacuum stability of scalar potentials is to check positive definiteness (or copositivity) of its coupling tensors, and such a coupling tensor is a 4th order and symmetric tensor. In this paper, we mainly discuss precise expressions of positive definiteness of 4th order tensors. More specifically, two analytically sufficient conditions of positive definiteness for 4th order 2 dimensional symmetric tensors are given by reducing orders of tensors, and applying these conclusions, some sufficient conditions for the positive definiteness of 4th order 3 dimensional symmetric tensors are derived. We also present several other sufficient conditions for the positive definiteness of 4th order 3 dimensional symmetric tensors. Finally, we test and verify the vacuum stability of general scalar potentials of two real singlet scalar fields and the Higgs boson by using these results.