Question

The values of $x$ for which the angle between the vectors $2x^2\hat{i}+4x\hat{j}+\hat{k}$ and $7\hat{i}-2\hat{j}+x\hat{k}$ are obtuse and the angle between the $z-$axis and $7\hat{i}-2\hat{j}+x\hat{k}$ is acute and less than $\frac{\pi }{6}$ is given by

• A. $x\in (0,\frac{1}{2})$
• B. $x>\frac{1}{2}$ or $x<0$
• C. $x\in (\frac{1}{2},15)$
• D. there is no such value for $x$.

Answer 1

The correct option is D there is no such value for $x$.

Let $\overrightarrow{a}=2x^2\hat{i}+4x\hat{j}+\hat{k}$ and $\overrightarrow{b}=7\hat{i}-2\hat{j}+x\hat{k}$.
The angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is obtuse
$\Rightarrow \overrightarrow{a}.\overrightarrow{b}<0$
$\Rightarrow 14x^2-8x+x<0$
$\Rightarrow x(2x-1)<0$
$\Rightarrow x\in (0,\frac{1}{2})$. ................$\left(1\right)$
Also it is given $\overrightarrow{b}.\hat{k}=x$ and $\frac{\overrightarrow{b}.\hat{k}}{\left|\overrightarrow{b}\right|}<\cos \frac{\pi }{6}=\frac{\sqrt{3}}{2}$
$\Rightarrow 2x>\sqrt{3}\sqrt{53+x^2}$
$\Rightarrow x^2>159$............................$\left(2\right)$
There is no common value for Eqs.$\left(1\right)$ and $\left(2\right)$