Question

*Exercise 6.3.6. Let ni, n2, nz, n, denote the outward pointing area vec- tors of a tetrahedron. Prove that their sum equals 0. (Hint: Let a1,...,a6 denote the edge vectors, write each area vector as a cross product of these, and apply the appropriate properties from Theorem 6.3.1 to the sum.)

Answer 1

Given n,,M2, 13 and my are outward pointing area. Vectors We have also mentioned the edge vectors. 7a, od A We 95 have to prove nt Matuzt ny=0 Cet in be the wren vector on the so y in terms of edge vectors represented as face OAB. can be o OBC n = 1 laxaz) Similarly considering me on face so = I l az xaz) Considering in on face OCA - 2 1 azxa) And considering my on face ABC ã, az, az OA, OB we can find as and ay in terms of since we are considering , az and az as and oc , then. as - CA so, in a triangle. - ÃO & CA + oc = o. - -a + as + az = 0 asa a, a

Similarly for ay vector. In DOAB AO of BA + OB = 0. where on = 0 BA = ay So . - 9 + au t ale o ay = a - az Sony = 1 1 , -as) x ( a - ar ) Summing all 4 vectors. taxas) + 4 (xai) +4 (az xaſ) + 16x6.) + 44 45 xu3 ) ulāxã) +4(- Ka - Rixã las yaz' ) = -7 lāsxas) So so out each & all the vectors cancel mit nap ng t ny =o other