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Vector Algebra

Question
CBSEENMA12034105
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Find λ when the scalar projection of straight a with rightwards arrow on top space equals space straight lambda straight i with hat on top space plus space straight j with hat on top space plus space 4 straight k with hat on top space space and space straight b with rightwards arrow on top space equals space 2 straight i with hat on top space plus space 6 straight j with hat on top space plus space 3 straight k with hat on top space is space 4 space units.

Solution

Here   straight a with rightwards arrow on top space equals space straight lambda space straight i with hat on top space plus space straight j with hat on top space plus space 4 space straight k with hat on top comma space space space straight b with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 6 space straight j with hat on top space plus space 3 space straight k with hat on top
therefore space space space open vertical bar straight b with rightwards arrow on top close vertical bar space equals space square root of 4 plus 36 plus 9 end root equals space square root of 49 space equals space 7
                straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space left parenthesis straight lambda right parenthesis space left parenthesis 2 right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis 6 right parenthesis space plus space left parenthesis 4 right parenthesis thin space left parenthesis 3 right parenthesis space equals space 2 space straight lambda space plus 6 plus 12 space equals space 2 straight lambda plus 18
Scalar projection of straight a with rightwards arrow on top space and space straight b with rightwards arrow on top = fraction numerator straight a with rightwards arrow on top. space straight b with rightwards arrow on top over denominator open vertical bar straight b with rightwards arrow on top close vertical bar end fraction
 therefore space space space 4 space equals space fraction numerator 2 space straight lambda space plus space 18 over denominator 7 end fraction space space space space space rightwards double arrow space space space space 28 space equals space 2 straight lambda plus 18 space space space space rightwards double arrow space space space 2 space straight lambda space equals space 10 therefore space space space space space straight lambda space equals space 5

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