Question

Let $a,b$ and $c$ be three vectors having magnitudes $1,1$ and $2$ respectively. If $a\times (a\times c)+b=0$, then the acute angle between $a$ and $c$ is

• A. $\frac{\pi }{4}$
• B. $\frac{\pi }{6}$
• C. $\frac{\pi }{3}$
• D. None of these

Answer 1

The correct option is B $\frac{\pi }{6}$

We have $a\times (a\times c)+b=0$

$\Rightarrow (a.c)a-(a.a)c=-b$

$\Rightarrow c-(a.c)a=b(\because a.a={\left|a\right|}^2=1)$

$\Rightarrow {|c-(a.c)a|}^2={\left|b\right|}^2$

$\Rightarrow {\left|c\right|}^2+{|a.c|}^2{\left|a\right|}^2-2(a.c)(a.c)={\left|b\right|}^2$

If $\theta$ is the angle between $a$ and $c$, we get

$4+{\left(2\cos \theta \right)}^2-2{\left(2\cos \theta \right)}^2=1$

$\Rightarrow 4{\cos }^2\theta =3\Rightarrow \cos \theta =\frac{\sqrt{3}}{2}$ $(\because \theta$ is acute $)$

$\Rightarrow \theta =\frac{\pi }{6}$