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?

• Correct answer
  A

  $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\overrightarrow{0}$
• B

  $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=$ unit vector
• C

  $\vec{a}+\vec{b}=\vec{c}$
• D

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

Solution

Position vectors of vertices A, B and 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 $\frac{1}{3}(\vec{a}+\vec{b}+\vec{c})=0$ $\vec{a}+\vec{b}+\vec{c}=0$

Hence, the correct answer is option: 'A'.