In this paper, the vector stochastic integrals are considered with respect to semimartingales. It establishes a result which can be easily applied to the problem derived from the different predictable process in the decomposition of semimartingales in the sense of vector stochastic integrals. As its applications, a straightforward proof of theorem 4.2 and theorem 4.6 in ref. is provided, and by using this idea, an important property of the vector stochastic integrals with respect to semimartingales is presented.