April 26th, 2017, 02:15 AM  #1 
Newbie Joined: Apr 2017 From: UK Posts: 1 Thanks: 0  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 pq Thanks in advance if anyone can help!! 
April 26th, 2017, 04:07 PM  #2 
Senior Member Joined: Aug 2012 Posts: 2,078 Thanks: 594 
$p \cdot (q + r) = q \cdot (p + r)$ $\implies p \cdot q + p \cdot r = q \cdot p + q \cdot r$ (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. Last edited by Maschke; April 26th, 2017 at 04:10 PM. 
April 26th, 2017, 05:15 PM  #3 
Senior Member Joined: Aug 2012 Posts: 2,078 Thanks: 594 
Typo  Last implication is supposed to be $p \cdot r = q \cdot r \implies (p  q) \cdot r = 0$. 

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