The P.V.'s of the vertices of a $∆$ABC are $\overline{i}+\overline{j}+\overline{k}, 4\overline{i}+\overline{j}+\overline{k}, \overline{4i}+5\overline{j}+\overline{k},$The P.V of the circumcentre of $∆$ ABC is

$$\begin{gathered} \overline{\mathrm{OA}}=\mathrm{i}+\mathrm{j}+\mathrm{k} , \overline{\mathrm{OB}}=4\mathrm{i}+\mathrm{j}+\mathrm{k} , \overline{\mathrm{OC}}=4\mathrm{i}+5\mathrm{j}+\mathrm{k} \\ \overline{\mathrm{AB}}=3\mathrm{i} \Rightarrow \left|\overline{\mathrm{AB}}\right|=3 \overline{) \overline{\mathrm{AC}}}=3\mathrm{i}+4\mathrm{j} \\ \overline{\mathrm{BC}}=4\mathrm{j} \Rightarrow \left|\overline{\mathrm{BC}}\right|=4 \overline{) \left|\overline{\mathrm{AC}}\right|}=5 \\ ∆\mathrm{ABC} \mathrm{is} \mathrm{right} \mathrm{angle} \mathrm{at} \mathrm{B} \\ ci\mathrm{rcum} \mathrm{centre} \mathrm{is} \mathrm{a} mid \mathrm{point} of a,c \\ =\left(\frac{5\overrightarrow{\mathrm{i}}+6\overrightarrow{\mathrm{j}}+2\overrightarrow{\mathrm{k}}}{2}\right) \end{gathered}$$