Question

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

Answer 1

Let, ABC be a given equilateral triangle and its vertices are A($\stackrel{\rightharpoonup }{a}$), B($\stackrel{\rightharpoonup }{b}$) and C($\stackrel{\rightharpoonup }{c}$).
Also, O($\stackrel{\rightharpoonup }{0}$) be the orthocentre of triangle ABC.
We know that centroid and orthocentre of equilateral triangle coincide at one point.
$$\begin{gathered} \mathrm{Orthocentre}\mathrm{of}△ABC=\stackrel{\rightharpoonup }{0} \\ \Rightarrow \mathrm{Centroid}△ABC=\stackrel{\rightharpoonup }{0} \\ \Rightarrow \frac{\stackrel{\rightharpoonup }{a}+\stackrel{\rightharpoonup }{b}+\stackrel{\rightharpoonup }{c}}{3}=\stackrel{\rightharpoonup }{0} \\ \therefore \stackrel{\rightharpoonup }{a}+\stackrel{\rightharpoonup }{b}+\stackrel{\rightharpoonup }{c}=\stackrel{\rightharpoonup }{0} \end{gathered}$$