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Vector Algebra

Question
CBSEENMA12034120
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The scalar product of the vector straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top with a unit vector along the sum of vectors 2 straight i with hat on top space plus space 4 space straight j with hat on top space minus space 5 straight k with hat on top and straight lambda straight i with hat on top space plus space 2 straight j with hat on top space plus space space 3 straight k with hat on top is equal to 1. Find the value of  λ.

Solution

Let straight a with rightwards arrow on top space equals space straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top
and straight b with rightwards arrow on top space equals space unit space vector space parallel space to space the space sum space of space vectors space 2 straight i with hat on top space plus space 4 straight j with hat on top space minus 5 straight k with hat on top comma space space straight lambda straight i with hat on top space plus space 2 space straight j with hat on top space plus space 3 space straight k with hat on top
         equals space fraction numerator left parenthesis 2 plus straight lambda right parenthesis space straight i with hat on top space plus space 6 space straight j with hat on top space minus space 2 space straight k with hat on top over denominator square root of left parenthesis 2 plus straight lambda right parenthesis squared plus left parenthesis 6 right parenthesis squared plus left parenthesis negative 2 right parenthesis squared end root end fraction space equals space fraction numerator 1 over denominator square root of straight lambda squared plus 4 straight lambda plus 44 end root end fraction space equals space open curly brackets left parenthesis 2 plus straight lambda right parenthesis straight i with hat on top space plus space space 6 space straight j with hat on top space minus space 2 space straight k with hat on top close curly brackets
Now,  straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 1
therefore space space space space space fraction numerator 1 over denominator square root of straight lambda squared plus 4 straight lambda plus 44 end root end fraction space left square bracket left parenthesis 1 right parenthesis plus space left parenthesis 2 plus straight lambda right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis 6 right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis negative 2 right parenthesis right square bracket space equals space 1 therefore space space space space space straight lambda plus 6 space equals square root of straight lambda squared plus 4 straight lambda plus 44 end root rightwards double arrow space space space space straight lambda squared plus 12 straight lambda space plus 36 space equals space straight lambda squared plus 4 straight lambda plus 44 rightwards double arrow space space space space space space space 8 straight lambda space equals space 8 space space space space space space rightwards double arrow space space space space space straight lambda space equals space 1 straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top 2 straight i with hat on top space plus space 4 straight j with hat on top space minus space 5 space straight k with hat on top space space space space space space space straight a with rightwards arrow on top space equals space straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top and space straight b with rightwards arrow on top space equals space unit space vector space parallel space to space the space sum space of space vectors space 2 straight i with hat on top space plus space 4 straight j with hat on top minus space 5 straight k with hat on top comma space straight lambda straight i with hat on top space plus space 2 straight j with hat on top space plus space 3 straight k with hat on top space space space space space space space space space space space space space space space equals space fraction numerator left parenthesis 2 plus straight lambda right parenthesis space straight i space plus space 6 space straight j with hat on top space minus space 2 space straight k with hat on top over denominator square root of left parenthesis 2 plus straight lambda squared right parenthesis plus left parenthesis 6 right parenthesis squared plus left parenthesis negative 2 right parenthesis squared end root end fraction space space space space space space space space space space space equals space fraction numerator 1 over denominator square root of straight lambda squared plus 4 straight lambda plus 44 end root end fraction open curly brackets left parenthesis 2 plus straight lambda right parenthesis space straight i with hat on top space plus space 6 space straight j with hat on top space minus space 2 space straight k with hat on top close curly brackets space Now space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 1 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 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 space space space space space space space space space space space space space space left parenthesis given right parenthesis therefore space space fraction numerator 1 over denominator square root of straight lambda squared plus 4 straight lambda plus 44 end root end fraction left square bracket left parenthesis 1 right parenthesis space plus space left parenthesis 2 plus straight lambda right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis 6 right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis negative 2 right parenthesis right square bracket space equals space 1 rightwards double arrow space space space space straight lambda squared plus 12 straight lambda plus 36 space equals space straight lambda squared plus 4 straight lambda plus 44 space rightwards double arrow space space space space 8 space straight lambda space equals space 8 space space space space space space space space space space space space space space space space space space space rightwards double arrow space space straight lambda space equals space 1

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