Manipal Entrance Test 2014 Math Paper

Let the first term and common ratio of a GP be α and β, then
a=α.βp−1,b=α.βq−1 and c=α.βr−1
∴ log‌a=log‌α+(p−1)‌log‌β
log‌b=log‌α+(q−1)‌log‌β and
log‌c=log‌α+(r−1)‌log‌β
The dot product of the given two vectors is
(q−r)‌log‌a+(r−p)‌log‌b+(p−q)‌log‌c
⇒ (q−r)[log‌α+(p−1)‌log‌β]+(r−p)
[log‌α+(q−1)‌log‌β]+(p−q)[log‌α+(r−1)‌log‌β]
⇒log‌α[q−r+r−p+p−q]+ log‌β[(p−1)(q−r)+(r−p)(q−1) +(r−1)(p−q)]
=0+0=0
⇒ The two vectors are perpedicular.