Question

$a, b, c$ are three vectors of magnitude, $\sqrt{3}, 1,2$ such that $a \times(a \times c)+3 b=O$. If $\theta$ is the angle between $a$ and $c$, then $\cos ^{2} \theta$ is equal to

• A. $\frac{1}{4}$
• B. $\frac{3}{4}$
• C. $\frac{1}{2}$
• D. none of these

Answer 1

Answer: (B)

We have, $a \times(a \times c)+3 b=0$
$\Rightarrow (a \cdot c) a-(a \cdot a) c+3 b=0$
$\Rightarrow (2 \sqrt{3} \cos \theta) a-3 c+3 b=0$
$\Rightarrow (2 \cos \theta) a-\sqrt{3} c+\sqrt{3} b=0$
$\Rightarrow |(2 \cos \theta) a-\sqrt{3} c|^{2}=|-\sqrt{3} b|^{2}$
$\Rightarrow 4 \cos ^{2} \theta|a|^{2}+3|c|^{2}-4 \sqrt{3} \cos \theta(a \cdot c)=3|b|^{2}$
$\Rightarrow 12 \cos ^{2} \theta+12-4 \sqrt{3} \cos \theta \times \sqrt{3} \times 2 \cos \theta=3$
$\Rightarrow 12 \cos ^{2} \theta+9-24 \cos ^{2} \theta=0$
$\Rightarrow 12 \cos ^{2} \theta=9 $
$\Rightarrow \cos ^{2} \theta=\frac{9}{12}=\frac{3}{4} .$