Adding a point to vector diagrams in multivariate analysis

A set of n base points P4i=1, 2,…, n), with known co-ordinates relative to orthogonal axes, and a further point Pn+1, with known distance from each of the base set, are given. The co-ordinates of Pn+1relative to the axes of the base set are found. The formula is particularly simple when the base set is referred to its principal axes, when the co-ordinates of Pn+1 for a subset of all the axes can be calculated from the co-ordinates of the Pi in this subset only. The classical results for adding a point to a principal components or canonical variates analyses are obtained when the base set is derived using the appropriate distance functions. An example is given.