I have 3 vectors in a high-dimensional euclidean space. I would like to express the length of their sum in terms of their lengths and inner products. For two vectors, I can do it with the law of cosines:
$c^2 = a^2 + b^2 - 2ab\cos\theta$
which in my context translates to:
$||c||^2 =||a||^2+||b||^2 -2 a \cdot b$
(where c = a+b and all are vectors, I project them into the span of a,b to get the triangle and then apply the law of cosines from trig)
How do I do this for 3 vectors, d=a+b+c?
I'll keep working on this and answer my own question if I come up with a solution.