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).

Answer 1

Given the vertices of the triangle $(1,-1,2),(2,1,3)$ and $(-1,2,-1)$. Then,
Position vectors are
$$\begin{gathered} \overrightarrow{a}=\hat{i}-\hat{j}+2\hat{k}. \\ \overrightarrow{b}=2\hat{i}+\hat{j}+3\hat{k}. \\ \overrightarrow{c}=-\hat{i}+2\hat{j}-\hat{k}. \end{gathered}$$
The centroid of a triangle is given by $\frac{\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}}{3}$
So, $\frac{\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}}{3}=\frac{\hat{i}-\hat{j}+2\hat{k}+2\hat{i}+\hat{j}+3\hat{k}-\hat{i}+2\hat{j}-\hat{k}}{3}=\frac{2\hat{i}+2\hat{j}+4\hat{k}}{3}=\frac{2}{3}\hat{i}+\frac{2}{3}\hat{j}+\frac{4}{3}\hat{k}$.