> > The goal is to find general equations for for all 26 Probability > functions Wi depending on the target vector Pta and other factors > because the solution is not unique. It is manifold! > > W_i=f(?,?,C1,C2,)
One simplification is to note that every solution consists of a "minimal vector" M of W values that meet the other conditions but have the sum of the W values less than 1, added to a "balanced vector" B of W values that make X = Y = Z = 0, where the sum of values in B is chosen so that the sum for B + M is 1. The possible B are easier to characterize. (Or if you just want one solution, take a simple choice of B, like W_(1,0,0) = W_(-1,0,0) with the rest of the W values zero.) As for the minimal vectors M, they will have all their W values zero, except for those "pointing the same way" as the target vector. This should make finding M much easier, since it cuts the number of variables needed.