How to prove the existence of vectors? 
I solved this problem for a few specific vectors but I don't know how to prove this in general?
Grateful for any kind of help!

Waving away degenerate cases, the four points $u_i$ either form a convex quadrilateral, or one of the $u_i$ lies in the interior of the triangle formed by the other three. In any case the point $u$ lies in a triangle formed by three of the four $u_i$.
This problem deals with an instance of Caratheodory's theorem on convex hulls of $> d+1$ points in ${\mathbb R}^d$.
