For 2 dimensional vectors: $V_1,V_2,...,V_n$
Suppose there is an exclusion vector $\vec{E}$.
In the process of adding these vectors (in any order), you have to check if the exclusion vector $E$ is reached. FOR Example: $[1,1], [2,2], [2,3]$ are my vectors. I have an exclusion vector $E = [3,4].$
There is a combination of adding these vectors that reaches the exclusion vector. $[1,1] +[2,3]+[2,2]$. Adding the first 2 yields $[3,4]$.
Now, of course, you could brute force all the combinations and check these.
But I am looking for an analytical method which can solve this problem. The input is $n$ vectors, and an exclusion $E$ and it can tell me if it is reached or not.
Somewhat more efficient than brute-force: Let $B_m$ be the set of vectors that can be obtained as the sum of a subset of $V_1, \ldots, V_m$. Compute iteratively using $B_0 = \{0\}$ and $B_{m+1} = B_m \cup \{V + V_m: V \in B_m\}$, until either you find $E$ or get to $B_n$ without finding it.
Alternatively: if $V_i = [a_i, b_i]$ and $E = [c,d]$, form the "generating function" $\prod_{i=1}^n (1 + x^{a_i} y^{b_i})$ and check if the coefficient of $x^c y^d$ is nonzero.
I don't think you can do much better than these. The one-dimensional version of your problem is the Subset Sum Problem, which is NP-complete.