How Do Perpendicular Vectors Around a Triangle Sum to Zero?

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SUMMARY

The discussion centers on proving that the sum of the perpendicular vectors nab, nbc, and nca around triangle ABC equals zero. Each vector is defined to have the same magnitude as its corresponding side vector (|nab|=|ab|, |nbc|=|bc|, |nca|=|ca|) and points outward from the triangle. The proof involves using properties of dot products, specifically that nab·ab=0, and applying a 90-degree rotation to the vectors. The conclusion drawn is that nab + nbc + nca = 0, demonstrating the balance of the vectors around the triangle.

PREREQUISITES
  • Understanding of vector notation and operations
  • Familiarity with dot product properties
  • Knowledge of triangle geometry and vector magnitudes
  • Basic concepts of rotation in a plane
NEXT STEPS
  • Study vector operations in depth, focusing on dot products and their geometric interpretations
  • Explore properties of perpendicular vectors in Euclidean geometry
  • Learn about circulation integrals and their applications in vector fields
  • Investigate rotation transformations in two-dimensional space
USEFUL FOR

This discussion is beneficial for students studying vector calculus, particularly those tackling proofs in geometry, as well as educators looking for examples of vector applications in triangle geometry.

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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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