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Vector Algebra

Question
CBSEENMA12035701
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The scalar product of the vector straight a with rightwards arrow on top equals space straight i with hat on top plus straight j with hat on top plus straight k with hat on top with a unit vector along the sum of vectors straight b with rightwards arrow on top equals 2 straight i with hat on top plus 4 straight j with hat on top minus 5 straight k with hat on top space space and space straight c with rightwards arrow on top space equals space straight lambda straight i with hat on top plus 2 straight j with hat on top plus 3 straight k with hat on top is equal to one. Find the value of straight lambda and hence find the unit vector along straight b with rightwards arrow on top plus straight c with rightwards arrow on top.

Solution

Given that
straight b with rightwards arrow on top space equals space 2 straight i with hat on top plus 4 straight j minus 5 straight k straight c with rightwards arrow on top equals straight lambda straight i with hat on top plus 2 straight j plus 3 straight k
Now consider the sum of vectors straight b with rightwards arrow on top plus straight c with rightwards arrow on top colon
straight b with rightwards arrow on top plus straight c with rightwards arrow on top equals space open parentheses 2 straight i with hat on top plus 4 straight j minus 5 straight k close parentheses plus open parentheses straight lambda straight i with hat on top plus 2 straight j plus 3 straight k close parentheses space space rightwards double arrow space space straight b with rightwards arrow on top plus straight c with rightwards arrow on top space equals space left parenthesis 2 plus straight lambda right parenthesis straight i with hat on top plus 6 straight j minus 2 straight k Let space straight n with hat on top space be space the space unit space vector space along space the space sum space of space vectors space straight b with rightwards arrow on top plus straight c with rightwards arrow on top colon straight n with hat on top space equals fraction numerator left parenthesis 2 plus straight lambda right parenthesis straight i with hat on top plus 6 straight j minus 2 straight k over denominator square root of left parenthesis 2 plus straight lambda right parenthesis squared plus 6 squared plus 2 squared end root end fraction The space scalar space product space of space straight a with rightwards arrow on top space and space straight n space is space 1. space Thus comma space straight a with rightwards arrow on top. straight n with hat on top space equals open parentheses straight i with hat on top plus straight j plus straight k close parentheses. space open parentheses fraction numerator left parenthesis 2 plus straight lambda right parenthesis straight i with hat on top plus 6 straight j minus 2 straight k over denominator square root of left parenthesis 2 plus straight lambda right parenthesis squared plus 6 squared plus 2 squared end root end fraction close parentheses rightwards double arrow 1 space equals fraction numerator 1 left parenthesis 2 plus straight lambda right parenthesis plus 1.6 minus 1.2 over denominator square root of left parenthesis 2 plus straight lambda right parenthesis squared plus 6 squared plus 2 squared end root end fraction
rightwards double arrow square root of left parenthesis 2 plus straight lambda right parenthesis squared plus 6 squared plus 2 squared end root space equals space 2 plus straight lambda plus 6 minus 2 rightwards double arrow square root of left parenthesis 2 plus straight lambda right parenthesis squared plus 6 squared plus 2 squared end root space equals straight lambda plus 6 rightwards double arrow left parenthesis 2 plus straight lambda right parenthesis squared plus 40 space equals space left parenthesis straight lambda plus 6 right parenthesis squared rightwards double arrow straight lambda squared plus 4 straight lambda plus 4 plus 40 equals straight lambda squared plus 12 straight lambda plus 36 rightwards double arrow 4 straight lambda plus 44 equals 12 straight lambda plus 36 rightwards double arrow 8 straight lambda equals 8 rightwards double arrow straight lambda equals 1
Thus, n is:
straight n equals fraction numerator left parenthesis 2 plus 1 right parenthesis straight i with hat on top plus 6 straight j minus 2 straight k over denominator square root of left parenthesis 2 plus 1 right parenthesis squared plus 6 squared plus 2 squared end root end fraction rightwards double arrow space straight n space equals space fraction numerator 3 straight i with hat on top plus 6 straight j minus 2 straight k over denominator square root of 3 squared plus 6 squared plus 2 squared end root end fraction rightwards double arrow straight n space equals fraction numerator 3 straight i with hat on top plus 6 straight j minus 2 straight k over denominator square root of 49 end fraction rightwards double arrow straight n space equals space fraction numerator 3 straight i with hat on top plus 6 straight j minus 2 straight k over denominator 7 end fraction rightwards double arrow straight n space equals 3 over 7 straight i with hat on top plus 6 over 7 straight j minus 2 over 7 straight k

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