If the p^th, q^th, r^th terms of a G.P are the positive numbers a, b, c then the angle between the vectors $\left({\mathrm{loga}}^2\right)\overline{\mathrm{i}} + \left({\mathrm{logb}}^2\right)\overline{\mathrm{j}} + \left({\mathrm{logc}}^2\right)\overline{\mathrm{k}}$  and  $(\mathrm{q} - \mathrm{r})\overline{\mathrm{i}} + (\mathrm{r} - \mathrm{p})\overline{\mathrm{j}} + (\mathrm{p}-\mathrm{q})\overline{\mathrm{k}}$  is

$$\begin{gathered} \mathrm{a}={\mathrm{AR}}^{\mathrm{p}-1} \mathrm{b}={\mathrm{AR}}^{\mathrm{q}-1} \mathrm{c}={\mathrm{AR}}^{\mathrm{r}-1} \\ \log \mathrm{a}=\log \mathrm{A}+(\mathrm{p}-1)\log \mathrm{R} \log \mathrm{b}=\log \mathrm{A}+(2-1)\log \mathrm{R} \log \mathrm{c}=\mathrm{logA}+(\mathrm{r}-1)\log \mathrm{R} \\ \overline{\mathrm{OA}}.\overline{\mathrm{OB}}=2 \log \mathrm{a}(\mathrm{q}-\mathrm{r})+2 \log \mathrm{b}(\mathrm{r}-\mathrm{p})+2 \log \mathrm{c} (\mathrm{p}-\mathrm{q}) \\ =2(\log \mathrm{A}+(\mathrm{p}-1)\log \mathrm{R}) (\mathrm{q}-\mathrm{r})+2(\log \mathrm{A}+(\mathrm{q}-1)\log \mathrm{R})(\mathrm{r}-\mathrm{p})+2(\log \mathrm{A}+(\mathrm{r}-1)\log \mathrm{R}) (\mathrm{p}-\mathrm{q}) \\ =0 \Rightarrow \mathrm{angle} ={90}^0 \end{gathered}$$