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Find the values of \(\alpha\) such that the vector \(\mathbf{a}=\hat{i}+3\hat{j}+\sin(2\alpha)\hat{k}\) forms an obtuse angle with the positive z-axis, and the vectors \(\mathbf{b}=(\tan\alpha)\hat{i}-\hat{j}+2\sin\left(\frac{\alpha}{2}\right)\hat{k}\) and \(\mathbf{c}=(\tan\alpha)\hat{i}+ (\tan\alpha)\hat{j}-3\sqrt{\csc\left(\frac{alpha}{2} ight)}\hat{k}\) are perpendicular to each other.

1. \(n\pi-\tan^{-1}2,\ n\in I\)
2. \((4n+2)\pi-\tan^{-1}2,\ n\in I\)
3. \(2n\pi-\tan^{-1}2,\ n\in I\)
4. \((2n+1)\pi-\tan^{-1}2,\ n\in I\)

Correct answer: \(2n\pi-\tan^{-1}2,\ n\in I\)

Solution

Perpendicularity of \(\mathbf{b}\) and \(\mathbf{c}\) gives a trigonometric equation that reduces to \(\tan\alpha=-2\). The obtuse angle condition with the positive z-axis requires the z-component of \(\mathbf{a}\) to be negative, which is satisfied by the same family of \(\alpha\) values. Hence \(\alpha=2n\pi-\tan^{-1}2\), \(n\in I\).

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