question_answer

If \[\vec{a},\text{ }\vec{b}\] 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?

A) \[\vec{a}+\vec{b}+\vec{c}=\vec{0}\]

B) \[\vec{a}+\vec{b}+\vec{c}=\,\,unit\,\,vector\]

C) \[\vec{a}+\vec{b}=\vec{c}\]

D) \[\vec{a}=\vec{b}+\vec{c}\]

Correct Answer: A

Solution :

[a] Position vectors of vertices A, B and C are \[\overset{\to }{\mathop{a}}\,,\overset{\to }{\mathop{b}}\,\] and \[\overset{\to }{\mathop{c}}\,\]. \[\because \] triangle is equilateral. \[\therefore \] Centroid and orthocenter will coincide. Centroid \[\equiv \] orthocenter position vector             \[=\frac{1}{3}(\overset{\to }{\mathop{a}}\,+\overset{\to }{\mathop{b}}\,+\overset{\to }{\mathop{c}}\,)\] \[\because \] given in question orthocenter is at origin. Hence \[\frac{1}{3}(\overset{\to }{\mathop{a}}\,+\overset{\to }{\mathop{b}}\,+\overset{\to }{\mathop{c}}\,)=0\] \[\overset{\to }{\mathop{a}}\,+\overset{\to }{\mathop{b}}\,+\overset{\to }{\mathop{c}}\,=0\]