In a triangle ABC, D, E and F are the mid-points of the sides BC, CA and AB respectively. Then \(\overrightarrow{AD}+\overrightarrow{BE}+\overrightarrow{CF}\) is equal to
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\vec{0}\)
Let \(\vec{a}, \vec{b}, \vec{c}\) be the position vectors of A, B, C. Each median goes from a vertex to the mid-point of the opposite side:
\( \displaystyle \overrightarrow{AD} = \frac{\vec{b}+\vec{c}}{2}-\vec{a} \)
\( \displaystyle \overrightarrow{BE} = \frac{\vec{c}+\vec{a}}{2}-\vec{b} \)
\( \displaystyle \overrightarrow{CF} = \frac{\vec{a}+\vec{b}}{2}-\vec{c} \)
On adding, each of \(\vec{a}, \vec{b}, \vec{c}\) appears twice with the factor \(\frac{1}{2}\) and once with the factor \(-1\):
\( \displaystyle \overrightarrow{AD}+\overrightarrow{BE}+\overrightarrow{CF} = (\vec{a}+\vec{b}+\vec{c})-(\vec{a}+\vec{b}+\vec{c}) = \vec{0} \)
Why the other options are incorrect
Option (1) \(\overrightarrow{AB}+\overrightarrow{AC}\) — Incorrect. This is \(2\,\overrightarrow{AD}\), twice the first median alone, which is not zero.
Option (2) \( \displaystyle \frac{3}{2}\,\overrightarrow{AC} \) — Incorrect. This is a non-zero vector, but the three medians cancel out completely.
Option (4) \(\overrightarrow{BC}\) — Incorrect. This is a non-zero side vector, but the three medians cancel out completely.