Is it possible that 3 vectors have all negative pairwise correlations?

Given three vectors a, b, and c, is it possible that correlations between a and b, a and c, and b and c are all negative? I.e. is this possible?



It is possible if the size of the vector is 3 or larger. For example


The correlations are

We can prove that for vectors of size 2 this is not possible:

The formula makes sense: if a1 is larger than a2, b2 has to be larger than b1 to make the correlation negative.

Similarly for correlations between (a,c) and (b,c) we get


Clearly, all of these three formulas can not hold at the same time.

Source : Link , Question Author : Antti A , Answer Author : Community

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