If the $${p^{th}},\,{q^{th}}$$   and $${r^{th}}$$ terms of a G.P. are positive numbers $$a,\,b$$  and $$c$$ respectively, then find the angle between the vectors $$\log \,{a^2}\hat i + \log \,{b^2}\hat j + \log \,{c^2}\hat k$$       and $$\left( {q - r} \right)\hat i + \left( {r - p} \right)\hat j + \left( {p - q} \right)\hat k$$

Solution :
Let $$A$$ be the first term and $$x$$ the common ratio of G.P.
So, $$a = A{x^{p - 1}} \Rightarrow \log \,a = \log \,A + \left( {p - 1} \right)\log \,x$$
Similarly, $$\log \,b = \log \,A + \left( {q - 1} \right)\log \,x$$
and $$\log \,c = \log \,A + \left( {r - 1} \right)\log \,x$$
If $$\overrightarrow \alpha = \log \,{a^2}\hat i + \log \,{b^2}\hat j + \log \,{c^2}\hat k$$
and $$\overrightarrow \beta = \left( {q - r} \right)\hat i + \left( {r - p} \right)\hat j + \left( {p - q} \right)\hat k\,{\text{then}}$$
$$\eqalign{ & \overrightarrow \alpha .\overrightarrow \beta = 2\left[ {\log \,a\left( {q - r} \right) + \log \,b\left( {r - p} \right) + \log \,c\left( {p - q} \right)} \right] \cr & = 2\left[ {\left( {q - r} \right)\left\{ {\log \,A + \left( {p - 1} \right)\log \,x} \right\} + \left( {r - p} \right)\left\{ {\log \,A + \left( {q - 1} \right)\log \,x} \right\} + \left( {p - q} \right)\left\{ {\log \,A + \left( {r - 1} \right)\log \,x} \right\}} \right] \cr & = 2\left[ {\left( {q - r + r - p + p - q} \right)\log \,A + \left( {qp - pr - p + r + pr - pq - r + p + pr - qr - p + q} \right)\log \,x} \right] \cr & = 0 \cr} $$
Hence, the angle between $$\overrightarrow \alpha $$ and $$\overrightarrow \beta $$ is $$\frac{\pi }{2}.$$