Question

If the angle between the vectors
$2 \alpha^{2} \hat{ i }+4 \alpha \hat{ j }+\hat{ k }$ and $7 \hat{ i }-2 \hat{ j }+\alpha \hat{ k }$ is obtuse, then

• A. $\alpha>\frac{1}{2}$
• B. $0<\alpha<\frac{1}{2}$
• C. $\alpha<0$
• D. $|\alpha|<\frac{1}{\text { ? }}$

Answer 1

Answer: (B)

Given vectors
$2 \alpha^{2} \hat{ i }+4 \alpha \hat{ j }+\hat{ k } \text { and } 7 \hat{ i }-2 \hat{ j }+\alpha \hat{ k }$
Angle between these vector are obtuse $\therefore \cos \theta<0$
$ \frac{\left(2 \alpha^{2} \hat{ i }+4 \alpha \hat{ j }+\hat{ k }\right)(7 \hat{ i }-2 \hat{ j }+\alpha \hat{ k })}{\left|2 \alpha^{2} \hat{ i }+4 \alpha \hat{ j }+\hat{ k }\right||7 \hat{ i }-2 \hat{ j }+\alpha \hat{ k }|} $
$ \Rightarrow 14 \alpha^{2}-8 \alpha+\alpha<0 \Rightarrow 14 \alpha^{2}-7 \alpha<0 $
$ \Rightarrow 7 \alpha(2 \alpha-1) < 0 \Rightarrow \alpha \in\left(0, \frac{1}{2}\right) $
$ \therefore 0 < \alpha < \frac{1}{2}$