If $D,E,F$ are respectively the mid point of $AB,\ AC$ and $BC$ in $\mathrm{\Delta}\ ABC$, then $\overrightarrow{\mathbf{BE}} + \overrightarrow{\mathbf{AF}}$ is equal to

Question

Dr. Abubacker · Accepted Answer

(a)
Let $O$ be the origin
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$	herefore\ \overrightarrow{\mathbf{BE}}\mathbf{+}\overrightarrow{\mathbf{AF}}\mathbf{=}\overrightarrow{\mathbf{OE}}\mathbf{-}\overrightarrow{\mathbf{OB}}\mathbf{+}\overrightarrow{\mathbf{OF}} - \overrightarrow{\mathbf{OA}}$
$= \frac{\overrightarrow{\mathbf{OA}}\mathbf{+}\overrightarrow{\mathbf{OC}}}{2} - \overrightarrow{\mathbf{OB}} + \frac{\overrightarrow{\mathbf{OB}}\mathbf{+}\overrightarrow{\mathbf{OC}}}{2} - \overrightarrow{\mathbf{OA}}$
$= \frac{\overrightarrow{\mathbf{OC}}}{2}\mathbf{+}\frac{\overrightarrow{\mathbf{OC}}}{2}\mathbf{+}\frac{\overrightarrow{\mathbf{OA}}}{2}\mathbf{-}\overrightarrow{\mathbf{OA}}\mathbf{+}\frac{\overrightarrow{\mathbf{OB}}}{2}\mathbf{-}\overrightarrow{\mathbf{OB}}$
$= \overrightarrow{\mathbf{OC}}\mathbf{-}\frac{\overrightarrow{\mathbf{OA}}\mathbf{+}\overrightarrow{\mathbf{OB}}}{2}\mathbf{=}\overrightarrow{\mathbf{OC}}\mathbf{-}\overrightarrow{\mathbf{OD}}\mathbf{=}\overrightarrow{\mathbf{DC}}$

Q. If $D,E,F$ are respectively the mid point of $AB,\ AC$ and $BC$ in $\mathrm{\Delta}\ ABC$, then $\overrightarrow{\mathbf{BE}} + \overrightarrow{\mathbf{AF}}$ is equal to

Explanation

Reviewed by Dr. Abubacker

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