My Math Forum Generalized Pythagorean theorem

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 October 4th, 2018, 10:51 PM #1 Member   Joined: Jun 2009 Posts: 83 Thanks: 1 Generalized Pythagorean theorem Hi, let S be bounded piece of a plane in the space E3 and let's note Si an orthogonal projection of S into xy, xz and yz planes respectively. Then it can be proved that (1) $\displaystyle area(S)^2=area(S1)^2+area(S2)^2+area(S3)^2$. But there is also a general theorem, that in a vector space with dot product where u,v,w are orthogonal vectors the identity (2) $\displaystyle |u+v+w|^2=|u|^2+|v|^2+|w|^2$ is true. There is great similarity between (1) and (2) here so my question is - can (1) be proved with help of (2), ie can S,Si be somehow interpreted as some vectors of some vector space (such that Si are orthogonal and S=S1+S2+S3)? Thank you for any suggestions. Last edited by skipjack; October 5th, 2018 at 12:13 AM.
 October 5th, 2018, 09:30 AM #2 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115 Interesting question. Any three vectors can be interpreted as orthogonal components of an area vector A=u+v+w. But that doesn't prove the area projection formula. You have to prove that the projection of an area vector A onto a plane whose normal is n is A.n. Not that easy as I recall, a little subtle.
 October 5th, 2018, 02:26 PM #3 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115 It’s interesting to note that you can project an area vector onto 3 non-orthogonal planes in which case A=u+v+w is still true but A^2=(u+v+w)^2 is u.u+u.v+u.w.......

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