The new orthogonal relationship is generalized for orthotropic elasticity of three-dimensions. The thought of how dual vectors are constructed in a new orthogonal relationship for theory of elasticity is generalized into orthotropic problems. A new dual vector is presented by the dual vector of the symplectic systematic methodology for elasticity that is over again sorted. A dual differential equation is directly obtained by using a mixed variables method. A dual differential matrix to be derived possesses a peculiarity of which principal diagonal sub-matrixes are zero matrixes. As a result of the peculiarity of the dual differential matrix, two independently and symmetrically orthogonal sub-relationships are discovered for orthotropic elasticity of three-dimensions. The dual differential equation is solved by a method of separation of variable. Based on the integral form of orthotropic elasticity a new orthogonal relationship is proved by using some identical equations. The new orthogonal relationship not only includes the symplectic orthogonal relationship but is also simpler. The physical significance of the new orthogonal relationship is the symmetry representation about an axis z for solutions of the dual equation. The symplectic orthogonal relationship is a generalized relationship but it may be appeared in a strong form with narrow sense in certain condition. This theoretical achievement will provide new effective tools for the research on analytical and finite element solutions to orthotropic elasticity of three-dimensions.