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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 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126 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 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126 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....... Tags dot product, generalized, pythagorean, pythagorean theorem, theorem, vector space Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post neelmodi Complex Analysis 1 March 19th, 2015 06:27 AM raul11 Number Theory 0 April 25th, 2014 04:36 PM johnny Geometry 10 September 20th, 2010 05:32 PM moonrains Geometry 2 January 7th, 2009 04:46 PM mohanned karkosh Geometry 1 October 22nd, 2007 06:13 AM

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