Question

If three vectors $a , b , c$ are such that $a \neq 0$ and $a \times b =2( a \times c ),| a |=| c |=1,| b |=4 $ and the angle between $b$ and $c$ is $\cos ^{-1}\left(\frac{1}{4}\right)$. Also $b -2 c =\lambda a$, then find the value of $\lambda$

• A. $ ~\pm \,4 $
• B. 14
• C. $ \pm \text{ }2~ $
• D. 12

Answer 1

Answer: (A)

$a \times b =2( a \times c )$
$\Rightarrow a \times( b -2 c )= 0$
$ \Rightarrow a$ is parallel to $b -2 c$
Now, $( b -2 c )=\lambda a $
$\Rightarrow | b -2 c |^{2}=\lambda^{2}| a |^{2}$
$\Rightarrow | b |^{2}+4| c |^{2}-4( b \cdot c )=\lambda^{2}| a |^{2}$
$\Rightarrow 16+4-4 \times| b ||c| \times \frac{1}{4}=\lambda^{2}$
$\Rightarrow 20-4=\lambda^{2} $
$\Rightarrow \lambda=\pm 4$