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Vectors — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsVectors3 Marks
Question
If $\mathrm{D}, \mathrm{E}, \mathrm{F}$ are the mid-points of the sides $\mathrm{BC}, \mathrm{CA}, \mathrm{AB}$ of a triangle $\mathrm{ABC}$, prove that $\overline{A D}+$

$\overline{B E}+\overline{C F}=0$

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Answer

Let $\bar{a}_t \bar{b}, \bar{c}_{,}, \bar{d}_{,}, \bar{e}, \bar{f}$ be the position vectors of the points $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}, \mathrm{F}$ respectively.

Since D, E, F are the midpoints of BC, CA, AB respec-tively, by the midpoint formula

$\bar{d}=\frac{\bar{b}+\bar{c}}{2}, \bar{e}=\frac{\bar{c}+\bar{a}}{2}, \bar{f}=\frac{\bar{a}+\bar{b}}{2}$

$\therefore \overline{\mathrm{AD}}+\overline{\mathrm{BE}}+\overline{\mathrm{CF}}=(\bar{d}-\bar{a})+(\bar{e}-\bar{b})+(\bar{f}-\bar{c})$

$=\left(\frac{\bar{b}+c}{2}-\bar{a}\right)+\left(\frac{\bar{c}+\bar{a}}{2}-\bar{b}\right)+\left(\frac{\bar{a}+\bar{b}}{2}-\bar{c}\right)$

$=\frac{1}{2} \bar{b}+\frac{1}{2} \bar{c}-\bar{a}+\frac{1}{2}-\bar{c}+\frac{1}{2} \bar{a}-\bar{b}+\frac{1}{2} \bar{a}+\frac{1}{2} \bar{b}-\bar{c}$

$=(\vec{a}+\bar{b}+\bar{c})-(\bar{a}+\bar{b}+\bar{c})=\overline{0}$

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