If `D ,\ E ,\ F` are the mid points of the side `B C ,\ C A` and `A B` respectively of a triangle ABC, write the value of ` vec A D+ vec B E+ vec C Fdot`

Explanation

Let the position vectors of ` A, B and C` be `vec{a}, vec{b} and vec{c}`

Then the position vector of `vec{d}, vec{e} and vec{f}` will be `frac{vec{b}+vec{c}}{2}, frac{vec{b}+vec{c}}{2} and frac{vec{b}+vec{a}}{2}` respectively.

Any vector `vec{A B}`, if the position vectors of point `A(vec{a})` and ` B(vec{b})` are known, can be written as `vec{b}-vec{a}`.

Then,

`vec{A D}+vec{B E}+vec{C F}=(vec{d}-vec{a})+(vec{e}-vec{b})+(vec{f}-vec{c})`

`=(frac{vec{b}+vec{c}}{2}-vec{a})+(frac{vec{a}+vec{c}}{2}-vec{b})+(frac{vec{b}+vec{a}}{2}-vec{c}) `

`=(vec{a}+vec{b}+vec{c})-(vec{a}+vec{b}+vec{c})=vec{0}`