How Do Perpendicular Vectors Around a Triangle Sum to Zero?

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The discussion centers on proving that the sum of the perpendicular vectors nab, nbc, and nca, which are directed outward from the sides of triangle ABC, equals zero. The vectors are defined to have the same magnitude as their corresponding sides, with nab perpendicular to ab, nbc to bc, and nca to ca. Participants suggest using the properties of dot products, noting that nab dot ab equals zero, and similar relationships for the other vectors. There is also mention of circulation integrals, indicating that the circulation around the triangle is zero, which may relate to the proof. The conversation emphasizes the geometric and algebraic relationships between the vectors and the triangle's sides.
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Homework Statement



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

Homework Equations



I'm guessing that the formula for the dot product will be used, and that nab(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.
 
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You know that

ab + bc + ca = 0

Now simply apply a rotation by 90 degrees
 

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