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.

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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