Explanation

(a)
Given that, $\overrightarrow{\mathbf{OA}} = 2\widehat{\mathbf{i}} + \widehat{\mathbf{j}} - \widehat{\mathbf{k}},\ \overrightarrow{\mathbf{OB}} = 3\widehat{\mathbf{i}} - 2\widehat{\mathbf{j}} + \widehat{\mathbf{k}}$ and $\overrightarrow{\mathbf{OC}} = \widehat{\mathbf{i}} + 4\widehat{\mathbf{j}} - 3\widehat{\mathbf{k}}$
$\overrightarrow{\mathbf{AB}}\mathbf{=}\overrightarrow{\mathbf{OB}}\mathbf{-}\overrightarrow{\mathbf{OA}}$
$= (3 - 2)\widehat{\mathbf{i}} + ( - 2 - 1)\widehat{\mathbf{j}} + (1 + 1)\widehat{\mathbf{k}}$
$= \widehat{\mathbf{i}} - 3\widehat{\mathbf{j}} + 2\widehat{\mathbf{k}}$
$|\overrightarrow{\mathbf{AB}}| = \sqrt{1^{2} + ( - 3)^{2} + 2^{2}}$
$= \sqrt{1 + 9 + 4} = \sqrt{14}$
$\overrightarrow{\mathbf{BC}}\mathbf{=}\overrightarrow{\mathbf{OC}}\mathbf{-}\overrightarrow{\mathbf{OB}}$
$= (1 - 3)\widehat{\mathbf{i}} + (4 + 2)\widehat{\mathbf{j}} + ( - 3 - 1)\widehat{\mathbf{k}}$
$= - 2\widehat{\mathbf{i}} + 6\widehat{\mathbf{j}} - 4\widehat{\mathbf{k}}$
$|\overrightarrow{\mathbf{BC}}| = \sqrt{( - 2)^{2} + 6^{2} + ( - 4)^{2}}$
$= \sqrt{4 + 36 + 16} = \sqrt{56}$
$\overrightarrow{\mathbf{CA}}\mathbf{=}\overrightarrow{\mathbf{OA}}\mathbf{-}\overrightarrow{\mathbf{OC}}$
$= (2 - 1)\widehat{\mathbf{i}} + (1 - 4)\widehat{\mathbf{j}} + ( - 1 + 3)\widehat{\mathbf{k}}$
$= \widehat{\mathbf{i}} - 3\widehat{\mathbf{j}} + 2\widehat{\mathbf{k}}$
$|\overrightarrow{\mathbf{CA}}| = \sqrt{1^{2} + ( - 3)^{2} + (2)^{2}}$
$= \sqrt{1 + 9 + 4} = \sqrt{14}$
It is clear that two sides of a triangle are equal.
$\therefore$ Points $A,B,C$ from an isosceles triangle.