| [ QuizWit ] in KIDS 글 쓴 이(By): guest (wiking) 날 짜 (Date): 1998년01월28일(수) 06시18분12초 ROK 제 목(Title): Re: [문제] 속빈 사면체의 무게 중심. 아이고.. 그럼 일단 사면체의 답을 쓰고, 제가 의미하는 바를 말씀드리지요. Choosing A, B, C, and D as vectors to vertices, we know that CM's of faces are a = (B+C+D)/3, etc. Fortunately, the area of bcd is proportional to the mass at a. And the same for other points. Now choosing the "CM" as the origin, Sum(MiRi) = 0. (A) multiplying (axb), we have... Mc*Vol(abcK) - Md*Vol(abdK) = 0. (here K denotes CM) (B) Fortunately, we know that for "Nae-Sim" of abcd, Vol(abcK) ~= Area(abc) ~= Md Vol(abdK) ~= Area(abd) ~= Mc s.t. (B) holds. Because that is valid for three independent combinations in axb, bxc.. etc, I proved that "Nae-Sim" of abcd is CM of ABCD. 젨PS> The Generalization of this proof needs the introduction of the following mapping on V^k in n dimensional vector space to R. d(R1^R2^..^Rn-k)(v1,..vk) = det(R1,R2,..Rn-k,v1,v2,..vk). Then, you will see that the previous posting is just a blind-minded generalization of this proof. |