Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

Let $\overline{\mathrm{a}},\overline{\mathrm{b}},\overline{\mathrm{c}}$ and $\overline{\mathrm{g}}$ be the position vectors of $\mathrm{A}$, $\mathrm{B},\mathrm{C}$ and $\mathrm{G}$ respectively. $\begin{array}{rl}& \overline{\mathrm{a}}=3\hat{\mathrm{i}}+1\hat{\mathrm{j}}+4\hat{\mathrm{k}},\\ & \overline{\mathrm{b}}=-4\hat{\mathrm{i}}+5\hat{\mathrm{j}}-3\hat{\mathrm{k}},\\ & \overline{\mathrm{g}}=-\hat{\mathrm{i}}+2\hat{\mathrm{j}}+\hat{\mathrm{k}},\end{array}$ $\mathrm{G}$ is centroid of $△\mathrm{ABC}$. $\begin{array}{rl}\therefore \bar{g}& =\frac{\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}}{3}\\ 3\bar{g}& =\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}\\ 3& 3(-\hat{\mathrm{i}}+2\hat{\mathrm{j}}+\hat{\mathrm{k}})=3\hat{\mathrm{i}}+\hat{\mathrm{j}}+4\hat{\mathrm{k}}-4\hat{\mathrm{i}}+5\hat{\mathrm{j}}-3\hat{\mathrm{k}}+\overline{\mathrm{c}}\\ \therefore \overline{\mathrm{c}}& =-3\hat{\mathrm{i}}+6\hat{\mathrm{j}}+3\hat{\mathrm{k}}-3\hat{\mathrm{i}}-\hat{\mathrm{j}}-4\hat{\mathrm{k}}+4\hat{\mathrm{i}}-5\hat{\mathrm{j}}+3\hat{\mathrm{k}}\\ & =-2\hat{\mathrm{i}}+0\hat{\mathrm{j}}+2\hat{\mathrm{k}}\end{array}$ $\therefore$ Third vertex $\mathrm{C}\equiv (-2,0,2)$