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Vector Algebra

Question
CBSEENMA12034104
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Find the projection of straight b with rightwards arrow on top plus straight c with rightwards arrow on top on straight a with rightwards arrow on top where
         straight a with rightwards arrow on top space equals space straight i with hat on top space plus space 2 space straight j with hat on top space plus space straight k with hat on top comma space space straight b with rightwards arrow on top space equals straight i with hat on top space plus space 3 space straight j with hat on top space plus space straight k with hat on top space and space straight c with rightwards arrow on top space equals space straight i with hat on top space plus space straight k with hat on top.

Solution
straight a with rightwards arrow on top space equals space straight i with hat on top space plus space 2 space straight j with hat on top space plus space straight k with hat on top comma space space space space space straight b with rightwards arrow on top space equals space straight i with hat on top space plus space 3 space straight j with hat on top space plus space straight k with hat on top comma space space space space straight c with rightwards arrow on top space equals space straight i with hat on top space plus space straight k with hat on top therefore space space space space space straight b with rightwards arrow on top space plus space straight c with rightwards arrow on top space equals space left parenthesis straight i with hat on top space plus space 3 space straight j with hat on top space plus space straight k with hat on top right parenthesis space plus space left parenthesis straight i with hat on top space plus space straight k with hat on top right parenthesis space equals space 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space space 2 space straight k with hat on top
Projection of left parenthesis straight b with rightwards arrow on top plus straight c with rightwards arrow on top right parenthesis space on space straight a with rightwards arrow on top
                         equals space fraction numerator left parenthesis straight b with rightwards arrow on top plus straight c with rightwards arrow on top right parenthesis space. space straight a with rightwards arrow on top over denominator open vertical bar straight a with rightwards arrow on top close vertical bar end fraction equals space fraction numerator left parenthesis 2 straight i with hat on top space plus space 3 straight j with hat on top space plus space 2 space straight k with hat on top right parenthesis. space space left parenthesis straight i with hat on top space plus space 2 space straight j with hat on top space plus space straight k with hat on top right parenthesis over denominator square root of left parenthesis 1 right parenthesis squared plus left parenthesis 2 right parenthesis space plus space left parenthesis 1 right parenthesis squared end root end fraction equals space fraction numerator left parenthesis 2 right parenthesis thin space left parenthesis 1 right parenthesis space plus space left parenthesis 3 right parenthesis thin space left parenthesis 2 right parenthesis space plus space left parenthesis 2 right parenthesis thin space left parenthesis 1 right parenthesis over denominator square root of 1 plus 4 plus 1 end root end fraction space equals fraction numerator 2 plus 6 plus 2 over denominator square root of 6 end fraction space equals fraction numerator 10 over denominator square root of 6 end fraction

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