I wasn't really understanding the sign influence for attractive and repulsive forces. Is an attractive force negative and repulsive force positive?

Yeah, these things always get confusing. It really comes down to establishing a convension and then sticking to that convension. In this case, your formula has a positive sign and force is proportional to the product of the two currents, so if the currents are in the same direction, you get a positve force (++=+ and --=+). Since we know that currents in the same direction will attract each other, then positive mean attraction. Currents in opposite directions are repulsive and give an overall negative sign because -+=- and +-=- .

However, you still need to be careful because the force in your coordinate system is another thing entirely and that plus/minus sign is not really relevant for the magnitude of the force, since magnitude is always positive. In this case, even though your force between L3 and L2 is repulsive (negative), L3 is being pushed in the positive directions for both (x) and (y) coordinates. So, in your coordinate system the force vector points in a positive direction for (x) and in a positive direction for (y).

I wish I could give you a simple rule, but I find it is best to think physically about the situation and double check directions and signs very carefully. Even after years of experience, this is often the place where mistakes will be made.

This is my calculations so far

so Total F(x) = 7.2x10^-4(N/m) - 4x10^-4(N/m)
and F(y) = 9.599x10^-4
seem ok?

then I would take the magnitude of F(x)+F(y) to fined the total force per unit length right?

I didn't check your numbers on a calculator, but the signs look correct and everything seems reasonable to me.

On your last statement there, I think I understand what you mean, but the formula is a little ambiguous. Some people might interpret F(x)+F(y) to be the addition of two scalars, but here it is the full vector with separate components. I would prefer to see it as Fx (x) +Fy (y) based on the notation system you seem to use to indicate coordinate directions. Actually, I don't like the use of (x) and (y) as symbols for unit vectors because it can be confused with functional dependence. For example. g=f(x) can mean g is a function of x, which is different than g=f1 (x)+ f2 (y) which could mean that g is a vector with f1 as the x-component and f2 as the y-component.

Anyway, I think you have it correct, but I just want to make sure that you are grasping the vector notation and the meaning of the symbols.

 

Yes, I see. I will take more care with my notation in future.
Thank you so much for you help