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April 26th, 2017, 01:15 AM   #1
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Little Vectors questions

Hi

Can someone help me with this question.

Vectors p q and r are such that p. (q+r) = q . (p+r)

Show that r must be perpendicular to p-q

Thanks in advance if anyone can help!!
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April 26th, 2017, 03:07 PM   #2
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$p \cdot (q + r) = q \cdot (p + r)$

$\implies p \cdot q + p \cdot r = q \cdot p + q \cdot r$ (by bilinearity).

Now if we are in a real vector space, which I suppose is assumed here, we have $p \cdot q = q \cdot p$. Note that in a complex vector space this does not hold, rather we have $p \cdot q = \overline{q \cdot p}$, the complex conjugate. Just mentioning this because commutativity of the dot product can not always be assumed.

However in this case we'll assume our vector space is over the real numbers so that we have $p \cdot r = q \cdot r \implies (p - q) \cdot r$ (by the bilinearity of the dot product) which shows that $p - q$ and $r$ are orthogonal.
Thanks from Country Boy

Last edited by Maschke; April 26th, 2017 at 03:10 PM.
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April 26th, 2017, 04:15 PM   #3
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Typo -- Last implication is supposed to be

$p \cdot r = q \cdot r \implies (p - q) \cdot r = 0$.
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