In a $∆$ ABC, P.V.'s of midpoints of AB, AC are $\overline{i}- \overline{j}+ \overline{k}, \overline{ j}+2\overline{k}$ . The P.V of centroid of $∆$ABC is $2\overline{i}+3\overline{ j} +4\overline{k}$. P.V of A is

$$\begin{gathered} \frac{\overline{\mathrm{OA}}+\overline{\mathrm{OB}}}{2}=\overrightarrow{\mathrm{i}}-\overrightarrow{\mathrm{j}}+\overrightarrow{\mathrm{k}}, \frac{\overline{\mathrm{OA}}+\overline{\mathrm{OC}}}{2}=\overrightarrow{\mathrm{j}}+2\overrightarrow{\mathrm{k}} \\ \overline{\mathrm{OA}}+\overline{\mathrm{OB}}=2\overrightarrow{\mathrm{i}}-2\overrightarrow{\mathrm{j}}+2\overrightarrow{\mathrm{k}}, \overline{\mathrm{OA}}+\overline{\mathrm{OC}}=2\overrightarrow{\mathrm{j}}+4\overrightarrow{\mathrm{k}} \\ 2\overline{\mathrm{OA}}+\overline{\mathrm{OB}}+\overline{\mathrm{OC}}=2\overrightarrow{\mathrm{i}}+6\overrightarrow{\mathrm{k}} \\ \mathrm{given} \frac{\overline{\mathrm{OA}}+\overline{\mathrm{OB}}+\overline{\mathrm{OC}}}{3}=2\overrightarrow{\mathrm{i}}+3\overrightarrow{\mathrm{j}}+4\overrightarrow{\mathrm{k}} \Rightarrow \overline{\mathrm{OA}}+\overline{\mathrm{OB}}+\overline{\mathrm{OC}}=6\overrightarrow{\mathrm{i}}+9\overrightarrow{\mathrm{j}}+12\overrightarrow{\mathrm{k}} \\ \overline{\mathrm{OA}}=2\overrightarrow{\mathrm{i}}+6\overrightarrow{\mathrm{k}}-6\overrightarrow{\mathrm{i}}-9\overrightarrow{\mathrm{j}}-12\overrightarrow{\mathrm{k}}=-4\overrightarrow{\mathrm{i}}-9\overrightarrow{\mathrm{j}}-6\overrightarrow{\mathrm{k}} \end{gathered}$$