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sum with vectors, convex polyhedron

Source: Polish MO Recond Round 1988 p6

September 9, 2024
vectorgeometry3D geometry

Problem Statement

Given is a convex polyhedron with k k faces S1,…,Sk S_1, \ldots, S_k . Let us denote the vector of length 1 perpendicular to the wall Si S_i (i=1,…,k i = 1, \ldots, k ) directed outside the given polyhedron by ni→ \overrightarrow{n_i} , and the surface area of this wall by Pi P_i . Prove that ∑i=1kPi⋅ni→=0→. \sum_{i=1}^k P_i \cdot \overrightarrow{n_i} = \overrightarrow{0}.
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