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 $x < 0$
$\Rightarrow x < 0$.