$\mathrm{If} {\mathrm{p}}^{\mathrm{th}},{\mathrm{q}}^{\mathrm{th}},{\mathrm{r}}^{\mathrm{th} }$ terms of G.P. are the positive numbers a, b, c then the angle between the vectors $\log a^3\overline{i}+\log b^3\overline{j}+\log c^3\overline{k}$
$\hspace{0.17em}\hspace{0.17em}and(q-r)\overline{i}+(r-p)\overline{j}+(p-q)\overline{k} is$

$$\begin{gathered} iet a_1 be first term and x be common ratio of terms in G.P \\ Now p^{th} term a=a_1{x}^{p-1}\Rightarrow \log a=\log a_1+(p-1)\log x \\ {q}^{th} term b=a_1{x}^{q-1}\Rightarrow \log b=\log a_1+(q-1)\log x \\ and r^{th} term c=a_1{x}^{r-1}\Rightarrow \log c=\log a_1+(r-1)\log x \\ Given vectors are \\ let \overline{a}=\log a^3\overline{i}+\mathrm{lo}g b^3\overline{j}+\log c^3\overline{k} \\ \Rightarrow \overline{a}=3\log a\overline{i}+3\log b\overline{j}+3\log c\overline{k} and \overline{b}=(q-r)\overline{i}+(r-p)\overline{j}+(p-q)\overline{k} \\ Now \overline{a}.\overline{b} =(3\log a\overline{i}+3\log b\overline{j}+3\log c\overline{k}).\left(\right(q-r)\overline{i}+(r-p)\overline{j}+(p-q\left)\overline{k}\right) \\ =3\log a(q-r)+3\log b(r-p)+3\log c(p-q) \\ \Rightarrow \overline{a}.\overline{b} \\ =3(q-r)(\log a_1+(p-1\left)\log x\right)+3(r-p)(\log a_1+(q-1\left)\log x\right)+3(p-q)(\log a_1+(r-1)\log x \\ =3\left[\begin{array}{cc}\log a_1(q-r+r-p+p-q)+\log x\left(\right(p-1\left)\right(q-r)+(q-1\left)\right(r-p)+(r-1\left)\right(p-q\left)\right)& \end{array}\right] \\ =0+0=0 \\ \therefore \overline{a}\cdot \overline{b}=0\Rightarrow \theta =\frac{\pi }{2} \end{gathered}$$