If two vertices of a triangle are \(\mathrm{A}(3,1,4)\) and \(\mathrm{B}(-4,5,-3)\) and the centroid of the triangle is \(\mathrm{G}(-1,2,1)\), then the third vertex \(\mathrm{C}\) of the triangle is

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.
\(\begin{aligned}
& \overline{\mathrm{a}}=3 \hat{\mathrm{i}}+1 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}, \\
& \overline{\mathrm{b}}=-4 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}, \\
& \overline{\mathrm{g}}=-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}},
\end{aligned}\)
\(\mathrm{G}\) is centroid of \(\triangle \mathrm{ABC}\).
\(\therefore \bar{g} =\frac{\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}}{3} \)
\( 3 \bar{g} =\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}} \)
\( 3 3(-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}})=3 \hat{\mathrm{i}}+\hat{\mathrm{j}}+4 \hat{\mathrm{k}}-4 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}+\overline{\mathrm{c}} \)
\( \therefore \overline{\mathrm{c}} =-3 \hat{\mathrm{i}}+6 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}-3 \hat{\mathrm{i}}-\hat{\mathrm{j}}-4 \hat{\mathrm{k}}+4 \hat{\mathrm{i}}-5 \hat{\mathrm{j}}+3 \hat{\mathrm{k}} \)
\( =-2 \hat{\mathrm{i}}+0 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}\)
\(\therefore\) Third vertex \(\mathrm{C} \equiv(-2,0,2)\)