Question

If the vectors $a=\left(c{\log }_{2}x\right)\hat{i}-6\hat{j}+3\hat{k}$ and $b=\left({\log }_{2}x\right)\hat{i}+2\hat{j}+\left(2c{\log }_{2}x\right)\hat{k}$ 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

Answer: (C)

$(-\frac{4}{3},0)$

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