The line of nodes en for the Euler angles (φ, ϑ, χ) is orthogonal to the plane of the two z axes e°z and ez. In the present paper we introduce two vectors, f°z and fz, in the plane of the z axes such that the set (f°z, en, fz) is biorthogonal to (e°z, en, ez). These new vectors are so easily visualized that their components can be written down by inspection of the figure illustrating the definition of the Euler angles, and thus the direction cosine e°r ⋅ es can be obtained simply. This derivation makes no use of matrices. The total angular momentum J for a system of particles, where the Euler angles are used as coordinates for overall rotation, is shown in a simple manner to be J = f°zpφ+enpϑ+fzpχ, where (pφ, pϑ, pχ) are the momenta conjugate to (φ, ϑ, χ).