Question

The value of $x$ for which the angle between the vectors $\overrightarrow{a}=x\hat{i}-3\hat{j}-\hat{k}$ and $\overrightarrow{b}=2x\hat{i}+x\hat{j}-\hat{k}$ s acute and the angle between the vectors $\overrightarrow{b}$ and the y-axis lies between $\frac{\pi }{2}$ and $\pi$ are

• A. 1, 2
• B. -2, -3
• C. < 0
• D. > 0

Answer 1

Answer: (C)

By the given conditions $\vec{a}.\vec{b}>0$ and $\vec{b}.\vec{j}<0$
Thus $2x^2-3x+1>0$ and $x<0$
$\Rightarrow \left(2x-1\right)\left(x-1\right)>0$ and $x<0$
$\Rightarrow 2x-1>0$ and $x-1>0$
or $2x-1<0$ and $x-1<0$ and $x<0$
$\Rightarrow \left[x>1\text{or}x<\frac{1}{2}\right]$ and $x<0$
$\Rightarrow x<0$.