The values of x for which the angle between the vectors $\vec a = x\hat i- 3\hat j-\hat k$ and $\vec b = 2x\hat i + x\hat j-\hat k$ is acute, and the angle between the vector $\vec b$ and the axis of ordinates is obtuse, are
Answer & explanation
Correct answer: option 3
Since the angle between the vectors $\vec a$ and $\vec b$ is acute and the angle between $\vec b$ and y-axis is obtuse. Therefore,
$\vec a.\vec b>0$ and $\vec b.\hat j<0$
$⇒2x^2-3x+1>0$ and $x <0$
$⇒(2x-1) (x-1) > 0$ and $x <0$
$⇒(x <1/2\, or\, x > 1)$ and $x <0⇒ x <0$