Question

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

• A. $0$
• B. $\frac{\pi }{2}$
• C. $\pi$
• D. $\frac{-\pi }{2}$

Answer 1

Answer: (A)

$0$

Let the G.P. be $\alpha ,\alpha \beta ,\alpha {\beta }^2,......$and $a=\alpha {\beta }^{p-1},b=\alpha {\beta }^{q-1},c=\alpha {\beta }^{r-1}$
$\log a=\log \alpha +(p-1)\log \beta$
$\log b=\log \alpha +(q-1)\log \beta$
$\log c=\log \alpha +(r-1)\log \beta$
$\overrightarrow{u}=\hat{i}\log a+\hat{j}\log b+\hat{k}\log c$
$=\hat{i}(\log \alpha +(p-1)\log \beta )+\hat{j}(\log \alpha +(q-1)\log \beta )+\hat{k}(\log \alpha +(r-1)\log \beta )$
$\overrightarrow{v}=(q-r)\hat{i}+(r-p)\hat{j}+(p-q)\hat{k}$
$\therefore \overrightarrow{u}.\overrightarrow{v}=\sum [\log \alpha +(p-1)\log \beta ](q-r)$
$=(\log \alpha -\log \beta )\sum (q-r)+\log \beta \sum p(q-r)$
$=(\log \alpha -\log \beta )[q-r+r-p+p-q]+\log \beta [p(q-r)+q(r-p)+r(p-q)]$
$=0+0=0$ on simplification