If vector a, vector b, vector c are the position vectors of the vertices of an equilateral triangle whose orthocentre is at the origin,

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 \(\vec a+\vec b+\vec c\).

Answer 1

Since in an equilateral triangle, orthocenter and centroid coincide, therefore the position vector of centroid is  \(\vec 0.\)

Also, the position vector of centroid G( \(\vec g\)) can be defined as \(\cfrac{\vec a+\vec b+\vec c}3\)

Therefore, \(\cfrac{\vec a+\vec b+\vec c}3\) = \(\vec 0\)  hence  \(\vec a+\vec b+\vec c=\vec 0\)