Question

The values of $x$ for which the angle between the vectors $\overrightarrow{a}=x\hat{i}-3\hat{j}-\hat{k}$ and $\overrightarrow{b}=2x\hat{i}+x\hat{j}-\hat{k}$ is acute, and the angle between the vectors $\overrightarrow{b}$ and the $y$-axis of ordinates is obtuse, are

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

Answer 1

The correct option is C All $x<0$

Since the angle between the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is acute and the angle between $\overrightarrow{b}$ and the $y$-axis is obtuse, therefore, $\overrightarrow{a}.\overrightarrow{b}>0$ and $\overrightarrow{b}.\hat{j}<0$

$\Rightarrow 2x^2-3x+1>0$ and $x<0$

$\Rightarrow (2x-1)(x-1)>0$ and $x<0$

$(x<\frac{1}{2}$ or $x>1)$ and $x<0\Rightarrow$ all $x<0$