If is an acute angle and the vector ( ) i +( ) j is perpendicular… - Vidyadip

MCQ

If $\theta$ is an acute angle and the vector $(\sin\theta)\hat{\text{i}}+(\cos\theta)\hat{\text{j}}$ is perpendicular to the vector $\hat{\text{i}}-\sqrt{3}\hat{\text{j}},$

A
$\frac{\pi}{6}$
B
$\frac{\pi}{5}$
C
$\frac{\pi}{4}$
✓
$\frac{\pi}{3}$

✓

Answer

Correct option: D.

$\frac{\pi}{3}$

The given vectors are perpendicular. so, their dot product is zero.

$\big[(\sin\theta)\hat{\text{i}}+(\cos\theta)\hat{\text{j}}\big].\big(\hat{\text{j}}-\sqrt{3}\hat{\text{j}}\big)=0$

$\Rightarrow\sin\theta-\sqrt{3}\cos\theta=0$

$\Rightarrow\sin\theta=\sqrt{3}\cos\theta$

$\Rightarrow\tan\theta=\sqrt{3}$

$\Rightarrow\theta=\frac{\pi}{3}$ (because $\theta$ is acute)

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