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Vector Algebra

Question
CBSEENMA12034182
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Find a unit vector perpendicular to the plane containing the vectors:
straight a with rightwards arrow on top space equals straight i with hat on top space plus space straight j with hat on top space minus space 2 straight k with hat on top space space and 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

Solution
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 space minus space 2 space straight k with hat on top comma space space space 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
straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space open vertical bar table row cell straight i with hat on top end cell cell straight j with hat on top end cell cell straight k with hat on top end cell row 1 1 cell negative 2 end cell row 1 cell negative 2 end cell 1 end table close vertical bar space equals space straight i with hat on top open vertical bar table row 1 cell negative 2 end cell row cell negative 2 end cell 1 end table close vertical bar space minus space straight j with hat on top space open vertical bar table row 1 cell negative 2 end cell row 1 1 end table close vertical bar space plus space straight k with hat on top space open vertical bar table row 1 1 row 1 cell negative 2 end cell end table close vertical bar space space space space space space space space space space space space equals space left parenthesis 1 minus 4 right parenthesis space straight i with hat on top space minus space left parenthesis 1 plus 2 right parenthesis space straight j with hat on top space plus space left parenthesis negative 2 minus 1 right parenthesis space straight k with hat on top space equals space minus 3 space straight i with hat on top space minus 3 space straight j with hat on top space minus space 3 space straight k with hat on top therefore space space space space open vertical bar straight a with rightwards arrow on top cross times space straight b with rightwards arrow on top close vertical bar space equals square root of 9 plus 9 plus 9 end root space equals square root of 27 space equals space 3 square root of 3 Unit space vector space perpendiuclar space to space straight a with rightwards arrow on top space and space straight b with rightwards arrow on top space equals plus-or-minus fraction numerator straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top over denominator open vertical bar straight a with rightwards arrow on top space cross times straight b with rightwards arrow on top space close vertical bar end fraction space space space space space space equals space plus-or-minus fraction numerator 1 over denominator 3 square root of 3 end fraction left parenthesis negative 3 space straight i with hat on top space minus space 3 space straight j with hat on top space minus space space 3 space straight k with hat on top right parenthesis space equals space plus-or-minus fraction numerator 1 over denominator square root of 3 end fraction left parenthesis straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top right parenthesis

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