If $\vec{a}, \vec{b}, \vec{c}$ are the position vectors of the vertices of an equilateral triangle whose circumcentre is at the origin, then $\vec{a} + \vec{b} + \vec{c}$ = _____________.

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Solution

Given:
$\vec{a}, \vec{b}, \vec{c}$ are the position vectors of the vertices of an equilateral triangle
circumcentre is at the origin

Since, the triangle is equilateral
Therefore, circumcentre = centroid

Let O be the centroid of the triangle.

$\vec{O O} = \frac{\vec{O A} + \vec{O B} + \vec{O C}}{3} \Rightarrow \vec{0} = \frac{\vec{O A} + \vec{O B} + \vec{O C}}{3} \Rightarrow \vec{0} = \frac{\vec{a} + \vec{b} + \vec{c}}{3} \Rightarrow \vec{0} = \vec{a} + \vec{b} + \vec{c}$

Hence, $\vec{a} + \vec{b} + \vec{c}$ = $\vec{0}$ .

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