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Vector Algebra

Question
CBSEENMA12033993
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If straight a with rightwards arrow on top space equals space straight i with hat on top space plus space 2 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 3 straight i with hat on top space plus space straight j with hat on top space minus space 5 space straight k with hat on top comma find a unit vector in the direction of straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top.

Solution

Here,          straight a with rightwards arrow on top space equals space straight i with hat on top space space plus space 2 straight j with hat on top space minus space straight k with hat on top comma space space straight b with rightwards arrow on top equals space 3 space straight i with hat on top space plus space straight j with hat on top space minus space 5 space straight k with hat on top
therefore space space space space straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top space equals space left parenthesis straight i with hat on top space plus space 2 space straight j with hat on top space minus space straight k with hat on top right parenthesis space minus space left parenthesis 3 space straight i with hat on top space plus space straight j with hat on top space minus space 5 space straight k with hat on top right parenthesis space equals space minus 2 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 therefore space space space space space space open vertical bar straight a with rightwards arrow on top minus straight b with rightwards arrow on top close vertical bar space equals space square root of left parenthesis negative 2 right parenthesis squared plus left parenthesis 1 right parenthesis squared plus left parenthesis 4 right parenthesis squared end root space equals space square root of 4 plus 1 plus 16 end root space equals space square root of 21 therefore space space space space space unit space vector space in space the space direction space of space straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top
                  equals space fraction numerator straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top over denominator open vertical bar straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top close vertical bar end fraction space equals space fraction numerator 1 over denominator square root of 21 end fraction space left parenthesis negative 2 space straight i with hat on top space plus space straight j with hat on top space space plus space 4 space straight k with hat on top right parenthesis equals space minus fraction numerator 2 over denominator square root of 21 end fraction straight i with hat on top space plus space fraction numerator 1 over denominator square root of 21 end fraction straight j with hat on top space plus space fraction numerator 4 over denominator square root of 21 end fraction straight k with hat on top.

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