Question

The values of $\alpha$ for which the vector $\vec{a}=\hat{i}+3 \hat{j}+\left(sin\,2\alpha\right)\hat{k}$
makes an obtuse angle with the z-axis and the vectors
$\vec{b}=\left(tan \, \alpha\right) \hat{i}-\hat{j}+2 \sqrt{Sin \,\frac{\alpha}{2}}\hat{k}$
$and \, \vec{c}=\left(tan\, \alpha\right)\hat{i}+\left(tan\,\alpha\right) \hat{j}-3 \sqrt{cos \, ec \frac{\alpha}{2}} \hat{k}$
are mutually orthogonal are

• A. $n \pi -tan^{-1} 2, n \in I$
• B. $\left(4n+2\right)\pi-tan^{-1} 2, n \in I$
• C. $2n\pi-tan^{-1} 2, n \in I$
• D. $\left(2n+1\right)\pi-tan^{-1} 2, n\in I$

Answer 1

Answer: (D)

$\vec{a}. \hat{k} <\,0 \Rightarrow \, sin \,2 \alpha <\, 0 \quad\dots\left(1\right)$
$\vec{b}\cdot\vec{c} \Rightarrow tan^{2}\, \alpha -tan \, \alpha-6=0 \Rightarrow tan \, \alpha =-2 \,or \, 3$
$sin\, 2\alpha<\,0 \, if \, tan\, \alpha = -2$
for which $sin \,\frac{\alpha}{2}>\,0 \, if \, \alpha$ lies in the second quadrant
$\therefore \, \alpha=\pi+tan^{-1}\left(-2\right)=\pi-tan^{-1}\,2$
General value is $\alpha=2 n\pi+\pi-tan^{-1} \, 2, n\in I$