Proving the Sum of Vector Areas in a Tetrahedron is Zero

[plexus0208]

Homework Statement

Four vectors are erected perpendicular to the four faces of a general tetrahedron. Each vector is pointing outwards and has a length equal to the area of the face. Show that the sum of these four vectors is zero.

Homework Equations

The Attempt at a Solution

Let A, B and C be vectors representing the three edges starting from a fixed vertex. Then, express each of the four vectors in terms of A, B and C.

 

[plexus0208]

I figured out how to do the problem.

But there's another part to the problem: Formulate and prove the analogous statement for a plane triangle.

What is meant by a "planar triangle"?

 

[lanedance]

i think it probably just means any normal triangle (normal in the sense that it is contained within in a plane in R^3, so planar)

can you show it for such? where I'm guessing the analogy is area to length

 

[LCKurtz]

Put your triangle in the xy plane; label the vertices A,B,C clockwise. Make vectors of the sides going from A to B, B to C, C to A (i.e, \vec a = B - A etc.). Those side vectors add to the zero vector. Now rotate the triangle 90 degrees counterclockwise and they become perpendicular to the original sides pointing outward and the right length. Presto!

 

[primoclark]

I am trying to solve this same problem. Would the person who asked the question and said he/she figured it out please tell me how its done? Unfortunately, it is just not clicking with me. Thank you.

 

[primoclark]

Could you explain how you solved this?