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Vector Algebra

Question
CBSEENMA12034052
Wired Faculty App

Prove that the vertices straight a with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space straight j with hat on top space minus space 2 space straight k with hat on top comma space space space straight b with rightwards arrow on top space equals space minus space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 4 space straight k with hat on top and straight c with rightwards arrow on top space equals space 4 space straight i with hat on top space minus space 2 space straight j with hat on top space minus space 6 space straight k with hat on top can form the sides of a triangle. Find the lengths of the medians of the triangle.

Solution
Let A, B, C be the points whose position vectors are
                                straight a with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space straight j with hat on top space minus space 2 space straight k with hat on top comma space straight b with rightwards arrow on top space equals space minus straight i with hat on top space plus space 3 space straight j with hat on top space plus space 4 space straight k with hat on top comma space space space straight c with rightwards arrow on top space equals space 4 space straight i with hat on top space minus space 2 space straight j with hat on top space minus space 6 space straight k with hat on top
   AB with rightwards arrow on top space equals space straight P. straight V. space of space straight B space minus space straight P. straight V. space of space straight A space equals space straight b with rightwards arrow on top minus straight a with rightwards arrow on top space equals space minus 4 space straight i with hat on top space plus 2 space straight j with hat on top space minus space 6 space straight k with hat on top BC with rightwards arrow on top space equals space straight P. straight V. space of space straight C space minus space straight P. straight V. space of space straight B space equals space straight c with rightwards arrow on top minus straight b with rightwards arrow on top space equals space 5 space straight i with hat on top space minus 5 space straight j with hat on top space minus space 10 space straight k with hat on top CA with rightwards arrow on top space equals space straight P. straight V. space of space straight A space minus space straight P. straight V. space of space straight C space equals space straight a with rightwards arrow on top space minus space straight c with rightwards arrow on top space equals space minus straight i with hat on top space plus space 3 space straight j with hat on top space plus space 4 space straight k with hat on top
therefore space space space AB with rightwards arrow on top space plus space BC with rightwards arrow on top space plus space CA with rightwards arrow on top space equals space 0 with rightwards arrow on top therefore space space space space ABC space is space straight a space triangle. space Let space straight D comma space straight E comma space straight F space be space mid minus points space of space sides space BC comma space CA comma space AB space respectively.

∴   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 respectively.
           AD with rightwards arrow on top space equals space straight P. straight V. space of space straight B space minus space straight P. straight V. space of space straight A 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 minus straight a with rightwards arrow on top
                 equals space 1 half left parenthesis straight b with rightwards arrow on top plus straight c with rightwards arrow on top minus 2 space straight a with rightwards arrow on top right parenthesis equals space 1 half left parenthesis negative stack straight i space with hat on top plus space 3 space straight j with hat on top space plus space 4 space straight k with hat on top space plus space 4 space straight i with hat on top space minus space 2 space straight j with hat on top space minus space 6 space straight k with hat on top space minus space space 6 space straight i with hat on top space minus space 2 space straight j with hat on top space plus space 4 space straight k with hat on top right parenthesis equals space 1 half left parenthesis negative 3 space straight i with hat on top space minus space straight j with hat on top space plus space 2 space straight k with hat on top right parenthesis
 BE with rightwards arrow on top space equals space straight P. straight V. space of space straight E space minus space straight P. straight V. space of space straight B space equals space fraction numerator straight c with rightwards arrow on top plus straight a with rightwards arrow on top over denominator 2 end fraction minus straight b with rightwards arrow on top space space space space space space space equals space 1 half left parenthesis straight c with rightwards arrow on top plus straight a with rightwards arrow on top space minus space 2 space straight b with rightwards arrow on top right parenthesis space space space space space space equals space 1 half left parenthesis 4 space straight i with hat on top space minus space 2 space straight j with hat on top space minus space 6 space straight k with hat on top space plus space 3 space straight i with hat on top space plus space straight j with hat on top space minus space 2 space straight k with hat on top space plus space 2 space straight j with hat on top space minus space 6 space straight j with hat on top space minus space 8 space straight k with hat on top right parenthesis space space space space space space equals space 1 half left parenthesis 9 space straight i with hat on top space minus space space 7 space straight j with hat on top space minus space 16 space straight k with hat on top right parenthesis
CF with rightwards arrow on top space equals space straight P. straight V. space of space straight F space minus space straight P. straight V. space of space straight C space equals space fraction numerator straight a with rightwards arrow on top plus straight b with rightwards arrow on top over denominator 2 end fraction space minus straight c with rightwards arrow on top space space space space space space equals space 1 half left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top minus 2 straight c with rightwards arrow on top right parenthesis space space space space space equals space 1 half left parenthesis 3 space straight i with hat on top space plus space straight j with hat on top space minus space 2 space straight k with hat on top space minus space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 4 space straight k with hat on top space minus space 8 space straight i with hat on top space plus space 4 space straight j with hat on top space plus space 12 space straight k with hat on top right parenthesis space space space space equals space 1 half left parenthesis negative 6 space straight i with hat on top space plus space 8 space straight j with hat on top space plus space 14 space straight k with hat on top right parenthesis space space space space equals space minus space 3 space straight i with hat on top space plus space 4 space straight j with hat on top space plus space 7 space straight k with hat on top
therefore space space space AD space equals space 1 half square root of 9 plus 1 plus 4 end root space equals 1 half square root of 14 space space space space space space space space BE space equals space 1 half square root of 81 plus 49 plus 256 end root space equals 1 half square root of 386 space space space space space space space space CF space equals space square root of 9 plus 16 plus 49 end root space equals square root of 74

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