Question

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

• A. $0<x<\frac{1}{2}$
• B. $\frac{1}{2}<x<15$
• C. $x>\frac{1}{2}$ or $x<0$
• D. none of these

Answer 1

The correct option is D none of these

The angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is obtuse.

$\therefore \overrightarrow{a}.\overrightarrow{b}<0\Rightarrow 14x^2-8x+x<0$

$\Rightarrow 7x(2x-1)<0\Rightarrow 0<x<\frac{1}{2}$

$\because$ The angle between $\overrightarrow{b}$ and the $z$-axis is acute and less than $\frac{\pi }{6}$

$\frac{\overrightarrow{b}.\overrightarrow{k}}{\left|\overrightarrow{b}\right|\left|\overrightarrow{k}\right|}>\cos \frac{\pi }{6}[\because \theta <\frac{\pi }{6}\Rightarrow \cos \theta >\cos \frac{\pi }{6}\Rightarrow \frac{\overrightarrow{b}.\overrightarrow{k}}{\left|\overrightarrow{b}\right|\left|\overrightarrow{k}\right|}>\cos \frac{\pi }{6}]$

$\Rightarrow \frac{x}{\sqrt{x^2+53}}>\frac{\sqrt{3}}{2}$

$\Rightarrow 4x^2>3x^2+159$

$\Rightarrow x^2>159$

$\Rightarrow x>\sqrt{159}$ or $x<-\sqrt{159}$