Question

Find the values of x for which the angle between the vectors $\overrightarrow{a}=x\hat{i}-3\hat{j}-1\hat{k}$ and $\overrightarrow{b}=2x\hat{i}+x\hat{j}-1\hat{k}$ is acute and the angle between the vector b and y-axis is obtuse.

• A. $x<0$
• B. $x>0$
• C. $x<\frac{1}{2}$
• D. $x>\frac{1}{2}$

Answer 1

The correct option is A $x<0$

Here, $a.b>0$ and $b.j<0$
$\therefore 2x^2-3x+1>0$ and $x<0$
or $(2x-1)(x-1)>0$ and $x<0$ $\therefore x<\frac{1}{2}$
or $x>1$, $x<\frac{1}{2}$ and $x<0$
Since $x<0,$ the value $x>1$ is ruled out.
$\therefore x<1/2$and $x<0\Rightarrow x<0$