question_answer

If D, E, F be the middle points of the sides BC, CA and AB of the triangle ABC, then \[\overrightarrow{AD}+\overrightarrow{BE}+\overrightarrow{CF}\] is

A) A zero vector         

B) A unit vector

C) 0             

D) None of these

Correct Answer: A

Solution :

\[\overrightarrow{AD}=\overrightarrow{OD}-\overrightarrow{OA}=\frac{\mathbf{b}+\mathbf{c}}{2}-\mathbf{a}=\frac{\mathbf{b}+\mathbf{c}-2\mathbf{a}}{2}\], (where \[O\] is the origin for reference) Similarly, \[\overrightarrow{BE}=\overrightarrow{OE}-\overrightarrow{OB}=\frac{\mathbf{c}+\mathbf{a}}{2}-\mathbf{b}=\frac{\mathbf{c}+\mathbf{a}-2\mathbf{b}}{2}\] and \[\overrightarrow{CF}=\frac{\mathbf{a}+\mathbf{b}-2\mathbf{c}}{2}\].      Now,     \[\overrightarrow{AD}+\overrightarrow{BE}+\overrightarrow{CF}\] \[=\frac{\mathbf{b}+\mathbf{c}-2\mathbf{a}}{2}+\frac{\mathbf{c}+\mathbf{a}-2\mathbf{b}}{2}+\frac{\mathbf{a}+\mathbf{b}-2\mathbf{c}}{2}=\mathbf{0}\].