The vector $\vec c$, directed along the internal bisector of the angle between the vectors $\vec c=7\hat i-4\hat j-4\hat k$ and $\vec b=-2\hat i-\hat j+2\hat k$ with $|\vec c|=5\sqrt{6}$, is
Answer & explanation
Correct answer: option 1
The required vector $\vec c$ is given by
$\vec c=λ(\hat a+\hat b)$
$⇒\vec c=λ\left(\frac{\vec a}{|\vec a|}+\frac{\vec b}{|\vec b|}\right)$
$⇒\vec c=λ\left\{\frac{1}{9}(7\hat i-4\hat j-4\hat k)+\frac{1}{3}(-2\hat i-\hat j+2\hat k)\right\}$
$⇒\vec c=\frac{λ}{9}(\hat i-7\hat j+2\hat k)$
$⇒|\vec c|=±\frac{λ}{9}\sqrt{1+49+4}=±\frac{λ}{9}\sqrt{54}$
But $|\vec c|=5\sqrt{6}$ [Given]
$⇒±\frac{λ}{9}\sqrt{54}=5\sqrt{6}⇒λ=±15$
Hence, $\vec c=±\frac{15}{9}(\hat i-7\hat j+2\hat k)=±\frac{5}{3}(\hat i-7\hat j+2\hat k)$