Question

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are the position vectors of the vertices of an equilateral triangle whose orthocentre is at the origin, then the centroid of the triangle satisfies which of the following relation?

• A. $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0$
• B. $\overrightarrow{a^2}=\overrightarrow{b^2}+\overrightarrow{c^2}$
• C. $\overrightarrow{a}+\overrightarrow{b}-\overrightarrow{c}=0$
• D. None of these

Answer 1

Answer: (A)

$\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0$

The position vector of the centroid of the triangle is $\frac{\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}}{3}$

Since the triangle is equilateral, therefore the orthocenter coincides with the centroid and hence $\frac{\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}}{3}=0$

$\therefore \overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0$