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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