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Vector Algebra

Question
CBSEENMA12034221
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Find the vectors area and also the scalar area of a triangle, the position vectors of whose vertices are negative straight i with hat on top plus straight j with hat on top plus 3 space straight k with hat on top comma space straight i with hat on top plus 2 space straight j with hat on top space plus space 4 space straight k with hat on top space and space 2 space straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top.

Solution

Let straight a with rightwards arrow on top space equals space minus straight i with hat on top plus straight j with hat on top plus 3 straight k with hat on top comma space space straight b with rightwards arrow on top space equals 3 space straight i with hat on top space plus space 2 space straight j with hat on top space plus space 4 space straight k with hat on top comma space space straight c with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top
be P.Vs. of A, B and C respectively.
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 minus straight i with hat on top minus 3 space straight j with hat on top space minus space 3 space straight k with hat on top BA 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 B space equals space straight a with rightwards arrow on top minus straight b with rightwards arrow on top space equals space minus 4 space straight i with hat on top space minus space straight j with hat on top space minus space straight k with hat on top space space space BC with rightwards arrow on top space cross times BA with rightwards arrow on top space equals space open vertical bar table row cell straight i with hat on top end cell cell space space straight j with hat on top end cell cell space space straight k with hat on top end cell row cell negative 1 end cell cell space minus 3 end cell cell space space minus 3 end cell row cell negative 4 space end cell cell negative 1 end cell cell negative 1 end cell end table close vertical bar space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals left parenthesis 3 minus 3 right parenthesis straight i with hat on top minus left parenthesis 1 minus 12 right parenthesis space straight j with hat on top space plus space left parenthesis 1 minus 12 right parenthesis space straight k with hat on top space equals space 11 space straight j with hat on top space minus space space 11 space straight k with hat on top
therefore space space vector space area space of space increment ABC space equals space 1 half space BC with rightwards arrow on top space cross times BA with rightwards arrow on top space equals space 1 half left parenthesis 11 space straight j with hat on top space minus space 11 space straight k with hat on top right parenthesis space equals space 11 over 2 left parenthesis straight j with hat on top minus straight k with hat on top right parenthesis
       Scalar area of increment ABC = open vertical bar 11 over 2 left parenthesis straight j with hat on top space minus space straight k with hat on top right parenthesis close vertical bar space equals space 11 over 2 square root of left parenthesis 1 right parenthesis squared plus left parenthesis negative 1 right parenthesis squared end root space equals space 11 over 2 square root of 2 space equals fraction numerator 11 over denominator square root of 2 end fraction

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