
Linear Algebra Linear Algebra Math Forum 
 LinkBack  Thread Tools  Display Modes 
October 4th, 2018, 09:51 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 4th, 2018 at 11:13 PM. 
October 5th, 2018, 08:30 AM  #2 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125 
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, 01:26 PM  #3 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125 
It’s interesting to note that you can project an area vector onto 3 nonorthogonal 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  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Application of Cauchy generalized integral theorem  neelmodi  Complex Analysis  1  March 19th, 2015 05:27 AM 
generalized gelfondschneide theorem  raul11  Number Theory  0  April 25th, 2014 03:36 PM 
Pythagorean theorem  johnny  Geometry  10  September 20th, 2010 04:32 PM 
pythagorean theorem  moonrains  Geometry  2  January 7th, 2009 03:46 PM 
new Pythagorean theorem  mohanned karkosh  Geometry  1  October 22nd, 2007 05:13 AM 