Question

$a,b,c$ are three vectors of magnitude, $\sqrt{3},1,2$ such that $a\times (a\times c)+3b=O$. 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, $a\times (a\times c)+3b=0$
$\Rightarrow (a\cdot c)a-(a\cdot a)c+3b=0$
$\Rightarrow (2\sqrt{3}\cos \theta )a-3c+3b=0$
$\Rightarrow (2\cos \theta )a-\sqrt{3}c+\sqrt{3}b=0$
$\Rightarrow |(2\cos \theta )a-\sqrt{3}c{|}^2=|-\sqrt{3}b{|}^2$
$\Rightarrow 4{\cos }^2\theta |a{|}^2+3|c{|}^2-4\sqrt{3}\cos \theta (a\cdot c)=3|b{|}^2$
$\Rightarrow 12{\cos }^2\theta +12-4\sqrt{3}\cos \theta \times \sqrt{3}\times 2\cos \theta =3$
$\Rightarrow 12{\cos }^2\theta +9-24{\cos }^2\theta =0$
$\Rightarrow 12{\cos }^2\theta =9$
$\Rightarrow {\cos }^2\theta =\frac{9}{12}=\frac{3}{4}.$