Question

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

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

Answer 1

Answer: (A)

The position vector of the centroid of the triangle is $\frac{\vec{ a }+\vec{ b }+\vec{ c }}{3}$. Since, the triangle is an equilateral, therefore the orthocentre coincides with the centroid and hence
$\frac{\vec{ a }+\vec{ b }+\vec{ c }}{3}=\vec{ 0 } \Rightarrow \vec{ a }+\vec{ b }+\vec{ c }=\vec{ 0 }$