In this paper, we study the general rogue wave solutions and their patterns in the vector (or M-component) nonlinear Schrödinger (NLS) equation. By applying the Kadomtsev-Petviashvili hierarchy reduction method, we derived an explicit solution for the rogue wave expressed by τ functions that are determinants of K×K block matrices (K=1,2,⋯,M) with an index jump of M+1. Patterns of the rogue waves for M=3,4 and K=1 are thoroughly investigated. We find that when a specific internal parameter is large enough, the wave patterns are linked to the root structures of generalized Wronskian-Hermite polynomial hierarchy in contrast with rogue wave patterns of the scalar NLS equation, the Manakov system and many others. Moreover, the generalized Wronskian-Hermite polynomial hierarchy includes the Yablonskii-Vorob\u27ev polynomial hierarchy and Okamoto polynomial hierarchies as special cases, which have been used to describe the rogue wave patterns of the scalar NLS equation and the Manakov system, respectively. As a result, we extend the most recent results by Yang {\it et al.} for the scalar NLS equation and the Manakov system. It is noted that the case M=3 displays a new feature different from the previous results. The predicted rogue wave patterns are compared with the ones of the true solutions for both cases of M=3,4. An excellent agreement is achieved
