Elementary treatments of a single angular momentum having components Jk (k =1, 2, 3) usually go as far as discussions of the spectra of these operators and the derivations of their standard, i.e., J2, J3-diagonal, representatives. On the other hand, the standard representatives of their eigenvectors are not usually obtained, leaving J3 aside, for which the problem is trivial. In fact, this part of the theory is most often dealt with only after a lengthy excursion into the theory of matrix representations. The representatives in question are obtained, in the first part of this paper, by means of essentially elementary arguments. Once these representatives are known, the standard representative of the unitary operator U which generates the state vector of a system after it has undergone a rotation R from the state vector prior to the rotation are deduced; this representative comprises just the familiar matrix representations of the rotation group. Finally, the relevance of harmonic oscillator theory to the present problem is briefly considered.