there are n * m possible sums. to find a solution, we can try to create such vectors such that all n * m sums are different.
Suppose the 1st vector is 1 2 3 ... n

Then the first element of next vector is taken to be n + 1(we require all unique).
Now we need to find the next element while making all n*p sums different, where p is the size of the 2nd vector.

n+1
1 2 3 ... n

initially all sums(n+2 to 2n+1) are distinct. Now we need to add new element such that
smallest possible new sum is greater than the previous sum
i.e. if new element is p
p+1>2n+1
or p >= 2n + 1
in general p+1>prev+n
p>=prev+n