A rigid transform of the E3 space into itself stands as the basis of a complete description of mechanical motions of solid bodies, where the orthogonal and normalized character (orthonormality) of the transform matrix are used as a postulate. However, the orthonormality is in fact a consequence of the condition of rigidity. The demonstration is outlined through the spectral analysis for application in the engineering of robots and astrodynamics. The problem posed is to directly find, in a general and confident manner, the elements of the rotation matrix when the direction and magnitude of the rotational displacement are given, which is an inverse design problem. Previously solved in 2-D, this problem is now extended to 3-D problems of mechanics and an implicit suggestion is made for the n-D mechanics.