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Vector Algebra

Question
CBSEENMA12034047
Wired Faculty App

If D, E and F are the mid-points of the sides of a triangle ABC, show that OA with rightwards arrow on top space plus space OB with rightwards arrow on top space plus space OC with rightwards arrow on top space equals space OD with rightwards arrow on top space plus space OE with rightwards arrow on top space plus space OF with rightwards arrow on top, where O is any arbitrary point.

Solution
Take O as origin.
Let straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top be position vectors of vertices A, B, C respectively.
                    therefore space space space space space space OA with rightwards arrow on top space equals space straight a with rightwards arrow on top comma space space space space space space space space space space OB with rightwards arrow on top space equals space straight b with rightwards arrow on top comma space OC with rightwards arrow on top space equals space straight c with rightwards arrow on top

Position vectors of D, E, F are
                               fraction numerator straight b with rightwards arrow on top plus straight c with rightwards arrow on top over denominator 2 end fraction comma space fraction numerator straight c with rightwards arrow on top plus straight a with rightwards arrow on top over denominator 2 end fraction comma space fraction numerator straight a with rightwards arrow on top plus straight b with rightwards arrow on top over denominator 2 end fraction space respectively.
therefore space space space space OD with rightwards arrow on top space equals space fraction numerator straight b with rightwards arrow on top plus straight c with rightwards arrow on top over denominator 2 end fraction comma space OE with rightwards arrow on top space equals fraction numerator straight c with rightwards arrow on top plus straight a with rightwards arrow on top over denominator 2 end fraction comma space OF with rightwards arrow on top space equals fraction numerator straight a with rightwards arrow on top plus straight b with rightwards arrow on top over denominator 2 end fraction space space space straight L. straight H. straight S. space equals space OA with rightwards arrow on top plus OB with rightwards arrow on top plus OC with rightwards arrow on top space equals space straight a with rightwards arrow on top plus straight b with rightwards arrow on top plus straight c with rightwards arrow on top space space space straight R. straight H. straight S. space equals space OD with rightwards arrow on top plus OE with rightwards arrow on top plus OF with rightwards arrow on top space space space space space space space space space space space space space space space space space equals space fraction numerator straight b with rightwards arrow on top plus straight c with rightwards arrow on top over denominator 2 end fraction plus fraction numerator straight c with rightwards arrow on top plus straight a with rightwards arrow on top over denominator 2 end fraction plus fraction numerator straight a with rightwards arrow on top plus straight b with rightwards arrow on top over denominator 2 end fraction space equals fraction numerator straight b with rightwards arrow on top plus straight c with rightwards arrow on top plus straight c with rightwards arrow on top plus straight a with rightwards arrow on top plus straight a with rightwards arrow on top plus straight b with rightwards arrow on top over denominator 2 end fraction space space space space space space space space space space space space space space space space space space space equals space fraction numerator 2 left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top plus straight c with rightwards arrow on top right parenthesis over denominator 2 end fraction space equals space straight a with rightwards arrow on top plus straight b with rightwards arrow on top plus straight c with rightwards arrow on top straight L. straight H. straight S. space equals space straight R. straight H. straight S.

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