I need to calculate
$(||2u+3v||^2 - ||2u-3v||^2)$
knowing that u and v are orthogonal.
I don't understand how i can calculate without any information on those vectors and why the fact that these vectors are orthogonal help to that.
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Andrei
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Witzig Adrien
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1 Answers
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Hint
For any vectors $x,y$, you have
$$\Vert x \Vert^2 - \Vert y\Vert^2 =\langle x-y , x+y \rangle$$
mathcounterexamples.net
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so x and y are vectors ? and the result is the scalar product of x-y and x+y ? – Witzig Adrien Mar 23 '20 at 19:49
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Just plug in $x=2u+3v$ – Andrei Mar 23 '20 at 19:50
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And $y=2u-3v$ to get the conclusion – mathcounterexamples.net Mar 23 '20 at 19:51
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@WitzigAdrien Yes. This is an identity satisfied by the inner product. – mathcounterexamples.net Mar 23 '20 at 19:52
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but how can i do the dot product of vectors i don't know anything ? – Witzig Adrien Mar 23 '20 at 19:55
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Have you tried to do the plug in that we suggested? – mathcounterexamples.net Mar 23 '20 at 19:55
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yes but i came to <6v.4u> how can i get further ? – Witzig Adrien Mar 23 '20 at 19:57
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1The inner product is bilinear... – mathcounterexamples.net Mar 23 '20 at 19:59
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1Also use the fact that the $u$ and $v$ vectors are perpendicular. That would simplify the scalar product – Andrei Mar 23 '20 at 20:02
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i think i understand, so (||2u+3v||^2 − ||2u−3v||^2) = 0 ? – Witzig Adrien Mar 23 '20 at 20:07
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That is right!! – mathcounterexamples.net Mar 23 '20 at 20:17