Question

The values of $\alpha$ for which the vector $\vec{a}=\hat{i}+3\hat{j}+\left(sin2\alpha \right)\hat{k}$
makes an obtuse angle with the z-axis and the vectors
$\vec{b}=\left(tan\alpha \right)\hat{i}-\hat{j}+2\sqrt{Sin\frac{\alpha }{2}}\hat{k}$
$and\vec{c}=\left(tan\alpha \right)\hat{i}+\left(tan\alpha \right)\hat{j}-3\sqrt{cosec\frac{\alpha }{2}}\hat{k}$
are mutually orthogonal are

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

Answer 1

Answer: (D)

$\vec{a}.\hat{k}<0\Rightarrow sin2\alpha <0\dots \left(1\right)$
$\vec{b}\cdot \vec{c}\Rightarrow ta{n}^2\alpha -tan\alpha -6=0\Rightarrow tan\alpha =-2or3$
$sin2\alpha <0iftan\alpha =-2$
for which $sin\frac{\alpha }{2}>0if\alpha$ lies in the second quadrant
$\therefore \alpha =\pi +ta{n}^{-1}\left(-2\right)=\pi -ta{n}^{-1}2$
General value is $\alpha =2n\pi +\pi -ta{n}^{-1}2,n\in I$