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Question
If θ is an acute angle and the vector (sin θ) \[\text{i}\] + (cos θ) \[\hat{j}\] is perpendicular to the vector \[\hat{i} - \sqrt{3} \hat{j} ,\] then θ =
Options
(a) \[\frac{\pi}{6}\]
(b) \[\frac{\pi}{5}\]
(c) \[\frac{\pi}{4}\]
(d) \[\frac{\pi}{3}\]
Solution
(d) \[\frac{\pi}{3}\]
\[\text{ The given vectors are perpendicular. So, their dot product is zero }.\]
\[\left[ \left( \sin \theta \right) \hat{i} + \left( \cos \theta \right) \hat{j} \right] . \left( \hat{i} - \sqrt{3} \hat{j} \right) = 0\]
\[ \Rightarrow \sin \theta - \sqrt{3} \cos \theta = 0\]
\[ \Rightarrow \sin \theta = \sqrt{3} \cos \theta\]
\[ \Rightarrow \tan \theta = \sqrt{3}\]
\[ \Rightarrow \theta = \frac{\pi}{3} (\text{ Because } \theta \text{ is acute })\]
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