Question

$a$ and $c$ are unit vectors and $|b|=4$. If angle between $b$ and $c$ is ${\cos }^{-1}\left(\frac{1}{4}\right)$ and $a\times b=2a\times c$, then $b=\lambda a+2c$, where $\lambda$ is equal to

• A. $\pm \frac{1}{4}$
• B. $\pm \frac{1}{2}$
• C. $\pm 4$
• D. None of the above

Answer 1

Answer: (C)

$\pm 4$

Given, $a\times b=2a\times c$
$\Rightarrow a\times b-2a\times c=0$
$\Rightarrow a\times (b-2c)=0$
$\Rightarrow a$ and $(b-c)$ are collinear.
$\Rightarrow b-2c=\lambda a$
$\Rightarrow |b-2c{|}^2={\lambda }^2|a{|}^2$
$\Rightarrow |b{|}^2+4|c{|}^2-4b\cdot c={\lambda }^2|a{|}^2$
$\Rightarrow 16+4-4\left(4\right)\left(1\right)\frac{1}{4}={\lambda }^2\left(1\right)$
Therefore, $\lambda =\pm 4$.