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 vector $\overrightarrow{b}$ and the axis of ordinates is obtuse, are

• A. $\{1,2\}$
• B. $\{-2,-3\}$
• C. $(-\infty ,0)$
• D. $(0,\infty )$

Answer 1

The correct option is C $(-\infty ,0)$

Given vectors:
$\overrightarrow{a}=x\hat{i}-3\hat{j}-\hat{k}$ and $\overrightarrow{b}=2x\hat{i}+x\hat{j}-\hat{k}$
for acute angle, $\overrightarrow{a}\cdot \overrightarrow{b}>0$
and for obtuse angle between axis of ordinates, $\overrightarrow{b}\cdot \overrightarrow{c}<0;$ here $\overrightarrow{c}=\hat{j}$
$\Rightarrow 2x^2-3x+1>0$ and $x<0$
$\Rightarrow (2x-1)(x-1)>0$ and $x<0$
$\Rightarrow x<\frac{1}{2}\text{or}x>1$ and $x<0$
taking common values : $x<0$