How can I prove that: 0<r^2<(r^2+(wl)^2)((1-w^2lc)^2+(wrc)^2) AA c>0 and all the other variables are bigger than zero using the scalar product of the vectors A=(r,wl) and B=(1-w^2 lc,wrc)?

Jackson Garner 2022-09-16 Answered
How can I prove that:
(1) 0 < r 2 < ( r 2 + ( w l ) 2 ) ( ( 1 w 2 l c ) 2 + ( w r c ) 2 )
c > 0 and all the other variables are bigger than zero using the scalar product of the vectors A = ( r , w l ) and B = ( 1 w 2 l c , w r c )?
I do not know how to get started on this problem and I do not see how I can use the scalar product of vectors to prove this inequality.
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Answers (1)

Simeon Hester
Answered 2022-09-17 Author has 15 answers
For r 0 , r 2 > 0 is trivial. The other inequality is ( A B ) 2 < ( A A ) ( B B ), which is Cauchy–Schwarz for non-parallel vectors (otherwise we could only claim ). In particular, check that r ( 1 w 2 l c ) + w l ( w r c ) = r. But A could be parallel to B, so < is in general incorrect.

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