In this paper, it is considered the exponential mappings g(t) = $e^{tA}$ Where t is a real parameter, g(t) is a dual orthogonal matrix and A is a dual anti-symmetric matrix. g(t) was written as a s arcparameter. Furthermore, applying the Gram-Schmidt method to the diffsrent vectors g'(s), g"(s), g"'(s), Frenet vectors, formulas and curvatures are calculated. In addition to, the parallel curves of g(s) in $ID^3$ have been obtained with help Frenet vectors of g(s).