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Question
Find the vector from the origin O to the centroid of the triangle whose vertices are (1, −1, 2), (2, 1, 3) and (−1, 2, −1).
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Solution
Given the vertices of the triangle \[(1, - 1, 2), (2, 1, 3)\] and \[( - 1, 2, - 1)\]. Then,
Position vectors are
\[\vec{a} = \hat{i} - \hat{j} + 2 \hat{k} . \]
\[ \vec{b} = 2 \hat{i} + \hat{j} + 3 \hat{k} . \]
\[ \vec{c} = - \hat{i} + 2 \hat{j} - \hat{k} .\]
The centroid of a triangle is given by \[\frac{\vec{a} + \vec{b} + \vec{c}}{3}\]
So,
\[\frac{\vec{a} + \vec{b} + \vec{c}}{3} = \frac{\hat{i} - \hat{j} + 2 \hat{k} + 2 \hat{i} + \hat{j} + 3 \hat{k} - \hat{i} + 2 \hat{j} - \hat{k}}{3} = \frac{2 \hat{i} + 2 \hat{j} + 4 \hat{k}}{3} = \frac{2}{3} \hat{i} + \frac{2}{3} \hat{j} + \frac{4}{3} \hat{k}\]
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