The uncertainty principle for vector-valued functions of $L^2({\mathbb{R}}^n,{\mathbb{R}}^m)$ with $n\ge 2$ are studied. We provide a stronger uncertainty principle than the existing one in literature when $m\ge 2$. The phase and the amplitude derivatives in the sense of the Fourier transform are considered when $m=1$. Based on these definitions, a generalized uncertainty principle is given.