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Vector Algebra

Question
CBSEENMA12034280
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If D, E, F are the mid points of the sides of triangle ABC, prove that
area left parenthesis increment DEF right parenthesis space equals space 1 fourth area space left parenthesis increment thin space ABC right parenthesis.

Solution
Take A as origin. Let position vectors of B and C be straight b with rightwards arrow on top space and space straight c with rightwards arrow on top respectively.
therefore  position vectors of D, E, F are 1 half left parenthesis straight b with rightwards arrow on top plus straight c with rightwards arrow on top right parenthesis comma space space space space 1 half straight c with rightwards arrow on top space and space space 1 half straight b with rightwards arrow on top space respectively.
therefore space space DE with rightwards arrow on top space equals space left parenthesis straight P. straight V. space of space straight E right parenthesis space minus space left parenthesis straight P. straight V. space of space straight D right parenthesis space space space space space space space space space space space space space equals space 1 half straight c with rightwards arrow on top space minus space 1 half left parenthesis straight b with rightwards arrow on top plus straight c with rightwards arrow on top right parenthesis space equals space minus 1 half straight b with rightwards arrow on top and space DF with rightwards arrow on top space equals space left parenthesis straight P. straight V. space of space straight F right parenthesis space space minus space left parenthesis straight P. straight V. space of space straight D right parenthesis space space space space space space space space space space space space space space equals space 1 half straight b with rightwards arrow on top space minus space 1 half left parenthesis straight b with rightwards arrow on top plus straight c with rightwards arrow on top right parenthesis space equals space minus 1 half straight c with rightwards arrow on top

therefore space space vector space area space of space increment DEF space equals space 1 half DE with rightwards arrow on top space cross times space DF with rightwards arrow on top space space equals space 1 half open parentheses negative 1 half straight b with rightwards arrow on top close parentheses space cross times space open parentheses negative 1 half straight c with rightwards arrow on top close parentheses space equals 1 fourth space open parentheses 1 half space left parenthesis straight b with rightwards arrow on top space cross times space stack straight c right parenthesis with rightwards arrow on top close parentheses space equals space 1 fourth left parenthesis vector space area space of space increment ABC right parenthesis.
Hence the result. 

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