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 = 

विकल्प

• (a) \[\frac{\pi}{6}\] 

• (b) \[\frac{\pi}{5}\] 

• (c)  \[\frac{\pi}{4}\] 

• (d)  \[\frac{\pi}{3}\]

Answer 1

(d) \[\frac{\pi}{3}\]

\[\text{ The given vectors are perpendicular. So, their dot product is zero }.\]
\[\left[ \left( \sin \theta \right) \hat{i} + \left( \cos \theta \right) \hat{j} \right] . \left( \hat{i} - \sqrt{3} \hat{j} \right) = 0\]
\[ \Rightarrow \sin \theta - \sqrt{3} \cos \theta = 0\]
\[ \Rightarrow \sin \theta = \sqrt{3} \cos \theta\]
\[ \Rightarrow \tan \theta = \sqrt{3}\]
\[ \Rightarrow \theta = \frac{\pi}{3} (\text{ Because } \theta \text{ is acute })\]