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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 write the value of \[\vec{a} + \vec{b} + \vec{c} .\]
Solution
Let, ABC be a given equilateral triangle and its vertices are A(\[\overrightarrow {a}\]), B([\overrightarrow {b}\]) , c(\[\overrightarrow {c}\]). Also,O(\[\overrightarrow {O}\]) be the orthocentre of triangle ABC.
We know that centroid and orthocentre of equilateral triangle coincide at one point.
\[\text{ Orthocentre of }\bigtriangleup ABC = \overrightarrow{O} \]
\[ \Rightarrow\text{ Centroid }\bigtriangleup ABC = \overrightarrow{O} \]
\[ \Rightarrow \frac{\overrightarrow{a} \ + \overrightarrow{b} + \overrightarrow{c}}{3} = \overrightarrow{o} \]
\[ \therefore \overrightarrow {a}+ \overrightarrow {b} + \overrightarrow {c} = \overrightarrow{0}\]
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