A vector C directed along the internal bisector  of the angle between the vectors  $\mathbf{a}=7\mathbf{i}-4\mathbf{j}-4\mathbf{k}$ and . $\mathbf{b}=-2\mathbf{i}-\mathbf{j}+2\mathbf{k},\text{ with }\left|\mathbf{c}\right|=5\sqrt{6}$ is 

The required vector $c$ is given by 

$\lambda \left(\frac{\mathbf{a}}{\left|\mathbf{a}\right|}+\frac{\mathbf{b}}{\left|\mathbf{b}\right|}\right)$

Now $\frac{\mathbf{a}}{\left|\mathbf{a}\right|}=\frac{1}{9}(7\mathbf{i}-4\mathbf{j}-4\mathbf{k})\text{ and }\frac{\mathbf{b}}{\left|\mathbf{b}\right|}=\frac{1}{3}(-2\mathbf{i}-\mathbf{j}+2\mathbf{k})$

$\Rightarrow \mathbf{c}=\lambda \left(\frac{1}{9}\mathbf{i}-\frac{7}{9}\mathbf{j}+\frac{2}{9}\mathbf{k}\right)\Rightarrow |\mathbf{c}{|}^2={\lambda }^2\times \frac{27}{81}\times 2$

Therefore,   ${\lambda }^2=225\text{ or }\lambda =15$.

$\Rightarrow \mathbf{c}=\frac{5}{3}(\mathbf{i}-7\mathbf{j}+2\mathbf{k})$.