Position vectors are much useful in several several fields, such as Differential Geometry, Mechanics and in Engineering, in particular in Dimensional Metrology. We generalize, for linear varieties in \( \mathbb R^n \), the corresponding results referred to the Euclidean ordinary space. The Moore-Penrose inverse of matrices plays an important rôle in this paper. Generalizations for three linear varieties of the Anderson-Duffin formulae are presented. We establish several formulae for a position vector of the intersection of linear varieties. Some characterization of the position vector is done in terms of centres of spheres. Results, in the context of commuting projections, are given as well.