This paper presents a fundamental consideration for the solution sets of linear optimal problems with a vector-valued performance index. A unified definition of three solution sets, which are efficient, proper and optimal, is given in terms of polar cones. Certain characteristics shared by the solution sets are investigated. The relationships between the solutions of the linear programming problems reduced from the vector-valued problem and the solution sets are established. The conditions which the solution sets should satisfy are derived by examining whether the intersection of two convex cones is empty or not. The conditions are useful for establishing an algorithm to find the solution sets. In order to facilitate understanding of the characteristics and the conditions obtained in this paper, a simple example is solved. A direction for further research of this problem and an application of the study are mentioned.