Are these two vectors coprime?

[phynewb]

Hi guys

I have a question about the coprime of two vectors
For two vectors (x1,x2) and (y1,y2).
Given a,b with gcf (a,b)=1 .i.e. relatively prime.
I do the linear combination of two vectors
a(x1,x2)+b(y1,y2)=n(z1,z2) with some common factor n and gcf(z1,z2)=1.
If n=1 for any a,b, two vectors are said co-prime.
I wonder if any criteria to prove two vectors are coprime.
For example, (2,3),(1,3) are not coprime b/c (2,3)+(1,3)=3(1,2).
But (7,3),(2,1) are coprime b/c a(7,3)+b(2,1)=(7a+2b,3a+b) and gcf(7a+2b,3a+b)=gcf(a,3a+b)=gcf(a,b)=1.
Also how to generalize it to vectors with n components?

Thank you

 

[Amir Livne]

I don't know if there's a name for this operation.
But I can recommend that you move the thread to the Number Theory forum, it seems to belong there more.