Let \(\mathrm{G}\) be the centroid of a triangle \(\mathrm{ABC}\) and \(0 \mathrm{be}\) any other point in that plane, then \(\overline{\mathrm{OA}}+\overline{\mathrm{OB}}+\overline{\mathrm{OC}}+\overline{\mathrm{OG}}=\)

(B)
Let \(\mathrm{O}\) be the origin.
Let \(\bar{a}, \bar{b}, \bar{c}\) be the position vectors of vertices \(A, B, C\) respectively.
\(\therefore \overline{\mathrm{OA}}+\overline{\mathrm{OB}}+\overline{\mathrm{OC}}=\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}\)
Given : \(G\) is the centroid of triangle
\(\begin{array}{l}
\therefore \overline{\mathrm{OG}}=\frac{\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}}{3} \\
\therefore \overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}=3 \overline{\mathrm{OG}} \\
\therefore \overline{\mathrm{OA}}+\overline{\mathrm{OB}}+\overline{\mathrm{OC}}+\overline{\mathrm{OG}}=3 \overline{\mathrm{OG}}+\overline{\mathrm{OG}}=4 \overline{\mathrm{OG}}
\end{array}\)