TITLE: Cool riddle involving integer vectors
QUESTION [6 upvotes]: Let $V$ be a $k$-dimensional vector ($k$ is given) of the form $$V=[n_1, n_2, ..., n_k],$$
where $n_1, n_2, ..., n_k$ are positive integers.
Your objective is to determine $V$ by choosing $T$ vectors of your choice, and receiving their dot products with $V$ respectively. 
for example, if my vector is $[1, 2]$, and your vectors are $[1, 1], [5, 3/2]$, you will get their dot poducts: $([1,1]*[1,2] =3, [5,3/2]*[1,2] = 8)$ 

What is the minimum number of vectors $T$ that you need to choose to determine $V$? (bonus: add the requirement that your vectors must consist of positive integers as well)

(p.s. there exists a solution $T<k$, it's not that obvious)
The solution to this riddle(s) is amazing, and i'll post it in a couple of days.

REPLY [3 votes]: Here is my answer:
Let $I = [ln(2),ln(3),...,ln(p_k)]$ where $p_k$ is the $k^{th}$ prime. 
$V*I=n_1ln(2)+n_2ln(3)+...+n_kln(p_k) = ln(2^{n_1}*3^{n_2}*...*p_k^{n_k})$
Therefore, by taking the prime factorizations of $e^{V*I}$, and looking at the exponents, we can deduce what $V$ is.
So by choosing $I$ and only $I$ to be our vector, we can fully deduce $V$ (and $minT=1$).