Question

If the vectors $\overrightarrow{a}=\left(clo{g}_{2}x\right)\hat{i}-6\hat{j}+2\hat{k}$ and $\overrightarrow{b}=\left(lo{g}_{2}x\right)\hat{i}+2\hat{j}+3\left(clo{g}_{2}x\right)\hat{k}$ make an obtuse angle for any $x\in (0,\infty )$ then c belongs to

• A. $(-\infty ,0)$
• B. $(-\infty ,-\frac{4}{3})$
• C. $(-\frac{4}{3},0)$
• D. $(-\frac{4}{3},\infty )$

Answer 1

The correct option is C $(-\frac{4}{3},0)$

For the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ to be inclined at an obtuse angle, we must have
$\overrightarrow{a}.\overrightarrow{b}<0$ for all $x\in (0,\infty )$
$\Rightarrow c(lo{g}_{2}x{)}^2-12+6c(lo{g}_{2}x)<0$ for all $x\in (0,\infty )$
$\Rightarrow c{y}^2+6cy-12<0$ for all $y\in R$, where $y=lo{g}_{2}x$
$\Rightarrow c<0$ and $\Rightarrow 36c^2+48c<0\Rightarrow c<0$and $c(3c+4)<0$
$\Rightarrow c<0$ and $-\frac{4}{3}<c<0$
$\Rightarrow c\in (-\frac{4}{3},0)$