If 7 hat{i}-4 hat{j}+5 hat{k} is the position | vedclass

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(C) Let $\vec{A} = 7 \hat{i}-4 \hat{j}+5 \hat{k}$ be the position vector of vertex $A$. Let $\vec{G}_{BCD} = \frac{\vec{B}+\vec{C}+\vec{D}}{3} = -\hat

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$\hat{i}+2 \hat{j}-\hat{k}$

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(C) Let $\vec{A} = 7 \hat{i}-4 \hat{j}+5 \hat{k}$ be the position vector of vertex $A$. Let $\vec{G}_{BCD} = \frac{\vec{B}+\vec{C}+\vec{D}}{3} = -\hat{i}+4 \hat{j}-3 \hat{k}$ be the position vector of the centroid of triangle $BCD$. The centroid $\vec{G}$ of the tetrahedron $ABCD$ is given by the formula: $\vec{G} = \frac{\vec{A}+\vec{B}+\vec{C}+\vec{D}}{4}$ We can rewrite this as: $\vec{G} = \frac{1}{4} [\vec{A} + 3(\frac{\vec{B}+\vec{C}+\vec{D}}{3})] = \frac{1}{4} [\vec{A} + 3 \vec{G}_{BCD}]$ Substituting the given values: $\vec{G} = \frac{1}{4} [(7 \hat{i}-4 \hat{j}+5 \hat{k}) + 3(-\hat{i}+4 \hat{j}-3 \hat{k})]$ $\vec{G} = \frac{1}{4} [7 \hat{i}-4 \hat{j}+5 \hat{k} - 3 \hat{i} + 12 \hat{j} - 9 \hat{k}]$ $\vec{G} = \frac{1}{4} [4 \hat{i} + 8 \hat{j} - 4 \hat{k}]$ $\vec{G} = \hat{i} + 2 \hat{j} - \hat{k}$

If $7 \hat{i}-4 \hat{j}+5 \hat{k}$ is the position vector of the vertex $A$ of a tetrahedron $ABCD$ and $-\hat{i}+4 \hat{j}-3 \hat{k}$ is the position vector of the centroid of the triangle $BCD$,then the position vector of the centroid of the tetrahedron $ABCD$ is

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