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
$\log {\mathrm{a}}^2\widehat{\mathrm{i}}+\log {\mathrm{b}}^2\widehat{\mathrm{j}}+\log {\mathrm{c}}^2\widehat{\mathrm{k}}\text{ and }(\mathrm{q}-\mathrm{r})\widehat{\mathrm{i}}+(\mathrm{r}-\mathrm{p})\widehat{\mathrm{j}}+(\mathrm{p}-\mathrm{q})\widehat{\mathrm{k}}$

Let A be the first term and X the common ratio of G.P.
So, $\mathrm{a}={\mathrm{Ax}}^{\mathrm{p}-1}\Rightarrow \log \mathrm{a}=\log \mathrm{A}+(\mathrm{p}-1)\log \mathrm{x}$
Similarly, $\log \mathrm{b}=\log \mathrm{A}+(\mathrm{q}-1)\log \mathrm{x}$
and $\log \mathrm{c}=\log \mathrm{A}+(\mathrm{r}-1)\log \mathrm{x}$
If $\overrightarrow{\mathrm{\alpha }}=\log {\mathrm{a}}^2\widehat{\mathrm{i}}+\log {\mathrm{b}}^2\widehat{\mathrm{j}}+\log {\mathrm{c}}^2\widehat{\mathrm{k}}$
and $\overrightarrow{\mathrm{\beta }}=(\mathrm{q}-\mathrm{r})\widehat{\mathrm{i}}+(\mathrm{r}-\mathrm{p})\widehat{\mathrm{j}}+(\mathrm{p}-\mathrm{q})\widehat{\mathrm{k}}$
$\begin{array}{l}\overrightarrow{\mathrm{\alpha }}.{\mathrm{\beta }}^{\to }=2[\log \mathrm{a}(\mathrm{q}-\mathrm{r})+\log \mathrm{b}(\mathrm{r}-\mathrm{p})+\log \mathrm{c}(\mathrm{p}-\mathrm{q}\left)\right]\\ =2\left[\right(\mathrm{q}-\mathrm{r}\left)\right\{\log \mathrm{A}+(\mathrm{p}-1)\log \mathrm{x}\}+(\mathrm{r}-\mathrm{p}\left)\right\{\log \mathrm{A}+(\mathrm{q}-1)\log \mathrm{x}\}\\ +(\mathrm{p}-\mathrm{q})\{\log \mathrm{A}+(\mathrm{r}-1)\log \mathrm{x}\}]\\ =2\left[\right(\mathrm{q}-\mathrm{r}+\mathrm{r}-\mathrm{p}+\mathrm{p})\log \mathrm{A}+(\mathrm{qp}-\mathrm{pr}-\mathrm{p}+\mathrm{r}+\mathrm{qr}-\mathrm{pq}-\mathrm{r}+\mathrm{p}+\mathrm{pr}-\mathrm{qr}-\mathrm{p}+\mathrm{q})\log \mathrm{x}]=0\end{array}$
Hence, the angle between $\overrightarrow{\mathrm{\alpha }} \mathrm{and} \overrightarrow{\mathrm{\beta }} \mathrm{is} \frac{\mathrm{\pi }}{2}$