Q247 / 270 · NDA 2022 (Phase 1) · Mathematics
If are the position vectors of the vertices $A, B, C$ respectively of a triangle $A B C$ and $G$ is the centroid of the triangle, then what is equal to?
- A$\frac{\vec{a}+\vec{b}+\vec{c}}{3}$
- B$\frac{2 \vec{a}-\vec{b}-\vec{c}}{3}$
- Correct answerC$\frac{\vec{b}+\vec{c}-2 \vec{a}}{3}$
- D$\frac{\vec{a}-2 \vec{b}-2 \vec{c}}{3}$
Solution
Given that $\vec{a}, \vec{b}$ and $\vec{c}$ are position vectors of the vertices $A, B$ and $C$ resp.
$\therefore$ position vector of centroid $G=\frac{\vec{a}+\vec{b}+\vec{c}}{3}$ Now, vector $\overrightarrow{\mathrm{AG}}=\frac{\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}}{3}-\overrightarrow{\mathrm{a}}$ $=\frac{\vec{b}+\vec{c}-2 \vec{a}}{3}$
Hence, the correct answer is option: 'C'.