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If $\vec{a}, \vec{b}$ and $\vec{c}$ are the position vectors of the vertices of an equilateral triangle whose orthocentre is at the origin, then which one of the following is correct?
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The correct answer is:
$\quad \vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$
Position vectors of vertices $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are $\vec{a}, \vec{b}$ and $\vec{c}$
$\because$ triangle is equilateral.
$\therefore$ Centroid and orthocenter will coincide. Centroid $\equiv$ orthocenter position vector
$=\frac{1}{3}(\vec{a}+\vec{b}+\vec{c})$
$\because$ given in question orthocenter is at origin.
Hence $\frac{1}{3}(\vec{a}+\vec{b}+\vec{c})=0$
$\vec{a}+\vec{b}+\vec{c}=0$
$\because$ triangle is equilateral.
$\therefore$ Centroid and orthocenter will coincide. Centroid $\equiv$ orthocenter position vector
$=\frac{1}{3}(\vec{a}+\vec{b}+\vec{c})$
$\because$ given in question orthocenter is at origin.
Hence $\frac{1}{3}(\vec{a}+\vec{b}+\vec{c})=0$
$\vec{a}+\vec{b}+\vec{c}=0$
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