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Vector Algebra

Question
CBSEENMA12034036
Wired Faculty App

If Q is the point of intersection of the medians of the triangle ABC, then prove that QA with rightwards arrow on top space plus space QB with rightwards arrow on top space plus space QC with rightwards arrow on top space equals space 0 with rightwards arrow on top.

Solution

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 the position vectors of vertices A, B, C respectively. Then position vector of centroid Q is fraction numerator straight a with rightwards arrow on top plus straight b with rightwards arrow on top plus straight c with rightwards arrow on top over denominator 3 end fraction.
L.H.S.  = QA with rightwards arrow on top space plus space OB with rightwards arrow on top space plus space QC with rightwards arrow on top
           equals space left parenthesis straight P. straight V. space of space straight A space minus space straight P. straight V. space of space straight Q right parenthesis space plus space left parenthesis straight P. straight V. space of space straight B space minus space straight P. straight V. space of space straight Q right parenthesis space plus space left parenthesis straight P. straight V. space of space straight C space minus space straight P. straight V. space of space straight Q right parenthesis
    equals space open parentheses straight a with rightwards arrow on top space minus space fraction numerator straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top plus straight c with rightwards arrow on top over denominator 3 end fraction close parentheses space plus space open parentheses straight b with rightwards arrow on top space minus space fraction numerator straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top space plus space straight c with rightwards arrow on top over denominator 3 end fraction close parentheses space plus space open parentheses straight c with rightwards arrow on top space minus space fraction numerator straight a with rightwards arrow on top plus straight b with rightwards arrow on top plus straight c with rightwards arrow on top over denominator 3 end fraction close parentheses equals space open parentheses straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top space plus space straight c with rightwards arrow on top close parentheses space minus space 3 space open parentheses fraction numerator straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top space plus space straight c with rightwards arrow on top over denominator 3 end fraction close parentheses space equals space left parenthesis straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top space plus space straight c with rightwards arrow on top right parenthesis space minus space left parenthesis straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top space plus space straight c with rightwards arrow on top right parenthesis space equals space 0 with rightwards arrow on top space equals space straight R. straight H. straight S.

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