I tried to solve a particular problem of mechanics and found some difficulties in the vector analysis part that I can't get rid of. It's probably some stupid mistake I made, but I can't see it now, and it doesn't appear something very obvious. I think it probably has something to do with complex numbers, but I couldn't find any that generates a negative root.
The problem appears at (2), because in ordinary cases the module of a vector is always positive. I.e, if $U=0\hat{i}+0\hat{j}+n\hat{k}$ is a vector, then it's module is $((U))=\sqrt[]{0^2+0^2+n^2}=\sqrt[]{n^2}$, which is positive since the power vanishes any negative sign. Now, $$\sqrt[]{n^2}<0 \Rightarrow n^2<0^2\Rightarrow n<0$$ since the product of two negative numbers is positive, we have $n.n>0$, a contradiction, since $n^2<0$. Anyway, does that mean the module of the vector is a complex number? If it does, why I found a negative one without imaginary parts or something. I know that people usually put negative modules to represent the direction of the force, i.e, $-6KN$ would mean the force contrary to the positive side with magnitude $6KN$; but that's just an abuse of language whose purpose is already done by the unities, modules having nothing to with it. In this case, we would have $6(-\hat{i})$.
Resuming the opera: if $$N_B=-80KN \Rightarrow ((N_B\hat{k}))=\sqrt[]{(-80)^2}=-80=\sqrt[]{(-80)(-80)}=\sqrt[]{80.80}=\sqrt[]{6400}$$ Any help is appreciated. Have a nice day.
P.S.: Sorry about the picture, I turned it on windows but it didn't do any good when I uploaded to SE.