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

$\frac{5}{3}(\hat i-7\hat j+2\hat k)$

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, $\vec c=±\frac{15}{9}(\hat i-7\hat j+2\hat k)=±\frac{5}{3}(\hat i-7\hat j+2\hat k)$