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Vector Algebra

Question
CBSEENMA12034183
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Find a unit vector perpendicular to each of the vector open parentheses straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top close parentheses and left parenthesis straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top right parenthesis, where 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 comma space space space space space straight b with rightwards arrow on top space equals space straight i with hat on top space plus space 2 space straight j with hat on top space plus space space 3 space straight k with hat on top.

Solution
straight a with rightwards arrow on top space equals straight i with hat on top plus straight j with hat on top plus straight k with hat on top comma space space space straight b with rightwards arrow on top space equals space straight i with hat on top space plus 2 straight j with hat on top space plus space 3 straight k with hat on top
therefore space space space space 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 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 4 space straight k with hat on top comma space space space straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top space equals space minus straight j with hat on top space minus space 2 space straight k with hat on top therefore space space space space space left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis space cross times space left parenthesis straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top right parenthesis space equals space straight i with hat on top open vertical bar table row 3 4 row cell negative 1 end cell cell negative 2 end cell end table close vertical bar minus space straight j with hat on top space open vertical bar table row 2 4 row 0 cell negative 2 end cell end table close vertical bar space plus space straight k with hat on top space open vertical bar table row 2 3 row 0 cell negative 1 end cell end table close vertical bar 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 equals space left parenthesis negative 6 plus 4 right parenthesis space straight i with hat on top space minus space left parenthesis negative 4 minus 0 right parenthesis space straight j with hat on top space plus space left parenthesis negative 2 minus 0 right parenthesis space straight k with hat on top space equals space minus 2 space straight i with hat on top space plus space 4 space straight j with hat on top space minus space 2 space straight k with hat on top
which is a vector perpendicular to both open vertical bar left parenthesis straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top right parenthesis space cross times space left parenthesis straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top right parenthesis close vertical bar space equals space square root of 4 plus 16 plus 4 end root space equals space square root of 24 space equals space square root of 4 space cross times 6 end root space equals space 2 square root of 6
therefore space space space required space unit space vector space space equals space fraction numerator left parenthesis straight a with rightwards arrow on top plus space straight b with rightwards arrow on top right parenthesis thin space cross times space left parenthesis straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top right parenthesis over denominator open vertical bar left parenthesis straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top right parenthesis space cross times space left parenthesis straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top right parenthesis close vertical bar end fraction 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 equals space fraction numerator 1 over denominator 2 square root of 6 end fraction left parenthesis negative 2 space straight i with hat on top space plus space 4 space straight j with hat on top space minus space 2 straight k with hat on top right parenthesis space equals space fraction numerator 1 over denominator square root of 6 end fraction left parenthesis negative 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

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