Question

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 $\log a^3\hat{i}+\log b^3\hat{j}+\log c^3\hat{k}$ and $(q-r)\hat{i}+(r-p)\hat{j}+(p-q)\hat{k}$

• A. $\frac{\pi }{6}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{3}$
• D. $\frac{\pi }{2}$

Answer 1

Answer: (D)

$\frac{\pi }{2}$

$\begin{array}{c}Accordingtoquestion.................\\ p^{th}term:a=m{n}^{p-1}suppose:firstterm=m\\ commonratioofG.P=n\\ b=m{n}^{q-1}\\ c=m{n}^{r-1}\\ firstvector:\\ \Rightarrow 3\log a\hat{i}+3\log b\hat{j}+3\log c\hat{k}\\ \overrightarrow{x}=3(\log m+(p-1\left)\log n\right)\hat{i}+3(logm+(q-1\left)logn\right)\hat{j}+3(\log m+(r-1\left)logn\right)\hat{k}\\ and,\\ \overrightarrow{y}=\left(q-r\right)\hat{i}+\left(r-p\right)\hat{j}+\left(p-q\right)\hat{k}\\ dotprodut:\\ \overrightarrow{x}.\overrightarrow{y}=3(\log m.q-r)+3\log n\left[(p-1)\left(q-r\right)\right]+3(logm.r-p)+3\log n\left[(q-1)\left(r-p\right)\right]+\\ +3(logm.p-q)+3\log n\left[(r-1)\left(p-q\right)\right]\\ =3\log m(q-r+r-p+p-q)+3\log n\left[(p-1)\left(q-r\right)+(q-1)\left(r-p\right)+(r-1)\left(p-q\right)\right]\\ =3\log m(q-r+r-p+p-q)+3\log n\left[pq-pr-q+r+qr-qp-r+p+rp-rq-p+q\right]\\ =3\log m\left(0\right)+3\log n\left[0\right]\\ =0+0=0\\ \therefore \overrightarrow{x}.\overrightarrow{y}=0\\ \overrightarrow{x}.\overrightarrow{y}=\left|x\right|\left|y\right|\cos \theta \\ \cos \theta =0\\ \therefore \theta =\frac{\pi }{2}\\ so,thatthecorrectoptionisD.\end{array}$