Question

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

• A. $\frac{\pi }{2}$
• B. $\frac{\pi }{3}$
• C. ${\sin }^{-1}\sqrt{a^2+b^2+c^2}$
• D. ${\sin }^{-1}\sqrt{p^2+q^2+r^2}$

Answer 1

The correct option is A $\frac{\pi }{2}$

Let first term of $G.P.$ be $x_0$ and the common ratio be $x$
$\Rightarrow a=x_0{x}^{p-1}\Rightarrow \log a=\log x_0+(p-1)\log x\cdots \left(i\right)$
Similarly
$\log b=\log x_0+(q-1)\log x\cdots \left(ii\right)$ and $\log c=\log x_0+(r-1)\log x\cdots \left(iii\right)$

Let $\overrightarrow{y}=\left(\log a^2\right)\hat{i}+\left(\log b^2\right)\hat{j}+\left(\log c^2\right)\hat{k}$ and $\overrightarrow{z}=(q-r)\hat{i}+(r-p)\hat{j}+(p-q)\hat{k}$
$\Rightarrow \overrightarrow{y}\cdot \overrightarrow{z}=2\log a(q-r)+2\log b(r-p)+2\log c(p-q)$
Using $\left(i\right),\left(ii\right)$ and $\left(iii\right),$ we have
$\Rightarrow \overrightarrow{y}\cdot \overrightarrow{z}=2(q-r)(\log x_0+(p-1\left)\log x\right)+2(r-p)(\log x_0+(q-1\left)\log x\right)+2(p-q)(\log x_0+(r-1\left)\log x\right)$
$\Rightarrow \overrightarrow{y}\cdot \overrightarrow{z}=2\log x_0(q-r+r-p+p-q)+2\log x(qp-pr-q+r+qr-pq-r+p+pr-qr-p+q)$
$\Rightarrow \overrightarrow{y}\cdot \overrightarrow{z}=0$
$\therefore$ Angle between two given vectors is $\frac{\pi }{2}$