Explanation

(a)
Let $\overrightarrow{\mathbf{OA}}\mathbf{=}\widehat{\mathbf{i}}\mathbf{+}\widehat{\mathbf{j}}\mathbf{+}\widehat{\mathbf{k}}\mathbf{,\ }\overrightarrow{\mathbf{OB}} = 5\widehat{\mathbf{i}} + 3\widehat{\mathbf{j}} - 3\widehat{\mathbf{k}}$ and $\overrightarrow{\mathbf{OC}} = 2\widehat{\mathbf{i}} + 5\widehat{\mathbf{j}} + 9\widehat{\mathbf{k}}$
$\therefore\ \overrightarrow{\mathbf{AB}} = 4\widehat{\mathbf{i}} + 2\widehat{\mathbf{j}} - 4\widehat{\mathbf{k}}\mathbf{,}\ \overrightarrow{\mathbf{BC}} = - 3\widehat{\mathbf{i}} + 2\widehat{\mathbf{j}} + 12\widehat{\mathbf{k}}$ and $\overrightarrow{\mathbf{AC}} = \widehat{\mathbf{i}} + 4\widehat{\mathbf{j}} + 8\widehat{\mathbf{k}}$
$\Rightarrow AB = 6,\ BC = \sqrt{157},\ AC = 9$
$\therefore$ Perimeter of $\mathrm{\Delta}\ ABC = 15 + \sqrt{157}$