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.