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bhargavi (67)

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 since   ,,are the  pth,Qth,Rth  terms  of   G.P      =arp-1   
 
=arQ-1       =arR-1    [R-1   in  power]   considering   the   two  given  vectors
as   A  &B   A.B=(Q-R)log3+(R-P)log3+(P-Q)log3
                         =3[(Q-R)l og+(R-P)log+(P-Q)log]
log=loga+logr(P-1);     log=loga+(Q-1)logr  ;          log  =loga+(R-1)logr
A.B=3[(Q-R)(loga+logr{P-1})+(R-P)(loga+(Q-1)logr+(P-Q)(loga+(R-1)logr)]
      =3[loga(Q-R+R-P+P-Q)+logr{(Q-R)(p-1)+(R-P)(Q-1)+(P-Q)(R-1)}]
     =3[0 +logr(0)]=0         
A.B=0     IMPLIES  the  angle    between   them  is    90   degrees
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