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Vector Algebra

Question
CBSEENMA12033998
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Find a vector of magnitude 5 units, and parallel to the resultant of the vectors 
straight a with rightwards arrow on top space equals space 2 straight i with hat on top space plus space 3 straight j with hat on top space minus space straight k with hat on top space and space straight b with rightwards arrow on top space equals space straight i with hat on top space minus space 2 space straight j with hat on top space plus space straight k with hat on top.

Solution

      straight a with rightwards arrow on top space equals space 2 straight i with hat on top plus 3 straight j with hat on top space minus straight k with hat on top comma space space space space space space space straight b with rightwards arrow on top space equals space straight i with hat on top space minus space 2 straight j with hat on top space plus space straight k with hat on top
therefore space space space space space straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top space equals space 3 straight i with hat on top space plus space straight j with hat on top therefore space space space space open vertical bar straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top close vertical bar space equals space square root of left parenthesis 3 right parenthesis squared plus left parenthesis 1 right parenthesis squared end root space equals space square root of 9 plus 1 end root space equals space square root of 10 therefore space space space straight a space unit space vector space parallel space to space straight a with rightwards arrow on top plus straight b with rightwards arrow on top space is space space equals space fraction numerator straight a with rightwards arrow on top plus straight b with rightwards arrow on top over denominator open vertical bar straight a with rightwards arrow on top space plus straight b with rightwards arrow on top close vertical bar end fraction space equals fraction numerator 1 over denominator square root of 10 end fraction left parenthesis 3 straight i with hat on top space plus space straight j with hat on top right parenthesis space equals space open parentheses fraction numerator 3 over denominator square root of 10 end fraction straight i with hat on top space plus space fraction numerator 1 over denominator square root of 10 end fraction straight j with hat on top close parentheses
Required vector  = 5 open parentheses fraction numerator straight a with rightwards arrow on top plus straight b with rightwards arrow on top over denominator open vertical bar straight a with rightwards arrow on top plus straight b with rightwards arrow on top close vertical bar end fraction close parentheses space equals space 5 space open parentheses fraction numerator 3 over denominator square root of 10 end fraction straight i with hat on top space plus space fraction numerator 1 over denominator square root of 10 end fraction straight j with hat on top close parentheses
                          equals space fraction numerator 15 over denominator square root of 10 end fraction straight i with hat on top space plus space fraction numerator 5 over denominator square root of 10 end fraction straight j with hat on top space equals space 3 over 2 square root of 10 space straight i with hat on top space plus space 1 half square root of 10 space straight j with hat on top

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