Q. If $\overrightarrow{\mathbf{a}},\overrightarrow{\mathbf{b}},\overrightarrow{\mathbf{c}}$ are the position vectors of the vertices of an equilateral triangle, whose orthocenter is at the origin, then
A) $\overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}} = \overrightarrow{\mathbf{0}}$
B) ${\overrightarrow{\mathbf{a}}}^{2} = {\overrightarrow{\mathbf{b}}}^{2} + {\overrightarrow{\mathbf{c}}}^{2}$
C) $\overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} = \overrightarrow{\mathbf{c}}$
D) None of these
Explanation
(a)
We know that the position vector of the centroid of the triangle is
$\frac{\overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}}}{3}\ \$
Since, the triangle is an equilateral, therefore the orthocentre coincides
With the centroid and hence,
$\frac{\overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}}}{3} = \overrightarrow{\mathbf{0}}$
$\Longrightarrow \ \overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}} = \overrightarrow{\mathbf{0}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.