This paper is concerned with the finite time blow-up phenomena for the vector nonlinear Schrdinger equations with a magnetic field which describe the spontaneous generation of a magnetic field in a cold plasma in the subsonic limit. After obtaining some a priori estimates,we prove under certain natural conditions that the solutions to the Cauchy problem of the vector nonlinear Schrdinger equations in two and three spatial dimensions blow up in a finite time. Assuming that a solution to the aforementioned vector nonlinear Schrdinger equations is radially symmetric with respect to spatial variables x,we show that if the initial energy is non-positive,then the solution blows up in three dimensions in a finite time. This paper is concerned with the finite time blow-up phenomena for the vector nonlinear Schrödinger equations with a magnetic field which describe the spontaneous generation of a magnetic field in a cold plasma in the subsonic limit. After obtaining some a priori estimates, we prove under certain natural conditions that the solutions to the Cauchy problem of the vector nonlinear Schrödinger equations in two and three spatial dimensions blow up in a finite time. Assuming that a solution to the vector of nonlinear Schrödinger equations is radially symmetric with respect to spatial variables x, we show that if the initial energy is non-positive, then the solution blows up in three dimensions in a finite time.