Generalized Pythagorean theorem

Jun 2009
83
1
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.
 
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Mar 2015
1,720
126
New Jersey
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.
 
Mar 2015
1,720
126
New Jersey
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.......