How Do Perpendicular Vectors Around a Triangle Sum to Zero?

[skate_nerd]

Homework Statement

So a, b, and c are points in the plane. Let n_ab, n_bc, and n_ca be vectors perpendicular to ab(vector), bc(vector), and ca(vector) respectively, and point towards the exterior of the triangle abc. Also, |n_ab|=|ab(vector)|, |n_bc|=|bc(vector)|, and |n_ca|=|ca(vector)|. Show that n_ab+n_bc+n_ca=0.

Homework Equations

I'm guessing that the formula for the dot product will be used, and that n_ab(dot)ab=0, and same for the other two vector combinations.

The Attempt at a Solution

Also, we have been learning about circulation integrals and line integrals. Not really sure if that proves much, but I know that the circulation around this triangle would be equal to 0 also, so there's something. Not sure really what else I have to go on though. I'm not very well versed in proofs, and my calc 3 teacher sure loves making us do them.

 

[dx]

You know that

ab + bc + ca = 0

Now simply apply a rotation by 90 degrees