# probability of randomly selected vector being orthogonal to atleast one vector from given list of random vectors in $n$ dimensional space

Let $$r = {[r_1, r_2, ..., r_n]}^T, r_i \in Z^+, r_i\le 1000000$$ ($$r$$ is uniformly randomly generated $$n-\text{dimensional}$$ vector with non-negative integer cordinates)

Let $$u = {[u_1, u_2, ..., u_n]}^T, u_i \in Z, -2\le u_i \le 2$$

Now, we have $$n^2$$ such vectors like $$u$$. Let's call list containing all of them $$L$$.

What is the probability (in terms of $$n$$) of $$r$$ being orthogonal to some vector $$v\in L$$?

• Each $u_i$ can take the $5$ possible values $-2,-1,0,1,2$. So number of such vectors $u$ will be $5^n$.
– QED
Sep 25, 2020 at 7:23
• Is the vound on $r_i$ really necessary?
– QED
Sep 25, 2020 at 7:25
• yes, becuase otherwise I suspect probability will be zero Sep 25, 2020 at 7:26
• yes all possible $u_i$ can be upto $5^n$ but we have chosen $n^2$ such $u$ to work with which makes a list $L$ Sep 25, 2020 at 7:27
• ALso if $r$ is orthogonal to $v\in L$ then so is $cr$ for any positive constant $c$. So if you bound $r_i$, then the denominator of the probability will be finite, but the numerator will be infinite.
– QED
Sep 25, 2020 at 7:30