Question 3
(GER) For each finite set $U$ of nonzero vectors in the plane we define $l(U)$ to be the length of the vector that is the sum of all vectors in $U$. Given a finite set $V$ of nonzero vectors in the plane, a subset $B$ of $V$ is said to be maximal if $l(B)$ is greater than or equal to $l(A)$ for each nonempty subset $A$ of $V$.
(a) Construct sets of $4$ and $5$ vectors that have $8$ and $10$ maximal subsets respectively.
(b) Show that for any set $V$ consisting of $n\ge1$ vectors, the number of maximal subsets is less than or equal to $2n$.
(a) Construct sets of $4$ and $5$ vectors that have $8$ and $10$ maximal subsets respectively.
(b) Show that for any set $V$ consisting of $n\ge1$ vectors, the number of maximal subsets is less than or equal to $2n$.