Question

If $a,b,c$ are $p^{th}$ and $q^{th}$ and $r^{th}$ terms of a $GP$, then the vectors $\log a\hat{i}+\log b\hat{j}+\log c\hat{k}$ and $(q-r)\hat{i}+(r-p)\hat{j}+(p-q)\hat{k}$ are

• A. Equal
• B. Parallel
• C. Perpendicular
• D. None of these

Answer 1

The correct option is C Perpendicular

Let the first term and common ratio of a GP be $\alpha$ and $\beta$, then
$a=\alpha \cdot {\beta }^{p-1},b=\alpha \cdot {\beta }^{q-1}$ and $c=\alpha \cdot {\beta }^{r-1}$
Therefore, $\log a=\log \alpha +(p-1)\log \beta$,
$\log b=\log \alpha +(q-1)\log \beta$ and
$\log c=\log \alpha +(r-1)\log \beta$
The dot product of the given two vectors is
$(q-r)\log a+(r-p)\log b+(p-q)\log c$
$\Rightarrow (q-r)[\log \alpha +(p-1\left)\log \beta \right]+(r-p0$
$[\log \alpha +(q-1\left)\log \beta \right]+(p-q)[\log \alpha +(r-1\left)\log \beta \right]$
$\Rightarrow \log \alpha [q-r+r-p+p-q]+\log \beta \left[\right(p-1\left)\right(q-r)+(r-p\left)\right(q-1)+(r-1\left)\right(p-q\left)\right]$
$=0+0=0$
Therefore, the two vectors are perpendicular.