Question

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

• A. A zero vector
• B. A unit vector
• C. 0
• D. None of these

Answer 1

Answer: (A)

A zero vector

Answer 2

$\overset{\rightarrow}{AD} = \overset{\rightarrow}{OD} - \overset{\rightarrow}{OA} = \frac{\mathbf{b} + \mathbf{c}}{2} - \mathbf{a} = \frac{\mathbf{b} + \mathbf{c} - 2\mathbf{a}}{2}$,

(where $O$ is the origin for reference)

Similarly, $\overset{\rightarrow}{BE} = \overset{\rightarrow}{OE} - \overset{\rightarrow}{OB} = \frac{\mathbf{c} + \mathbf{a}}{2} - \mathbf{b} = \frac{\mathbf{c} + \mathbf{a} - 2\mathbf{b}}{2}$ and

$\overset{\rightarrow}{CF} = \frac{\mathbf{a} + \mathbf{b} - 2\mathbf{c}}{2}$.

Now, $\overset{\rightarrow}{AD} + \overset{\rightarrow}{BE} + \overset{\rightarrow}{CF}$

$= \frac{\mathbf{b} + \mathbf{c} - 2\mathbf{a}}{2} + \frac{\mathbf{c} + \mathbf{a} - 2\mathbf{b}}{2} + \frac{\mathbf{a} + \mathbf{b} - 2\mathbf{c}}{2} = \mathbf{0}$.