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Vector Algebra

Question
CBSEENMA12034172
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Find the unit vector perpendicular to the vectors 
straight a with rightwards arrow on top space equals space 4 space straight i with hat on top space plus space 3 space 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 2 space straight i with hat on top space minus space 3 space stack straight j space with hat on top space minus space 3 space straight k with hat on top. 

Solution
straight a with rightwards arrow on top space equals space 4 space straight i with hat on top space plus space 3 space straight j with hat on top space minus space straight k with hat on top comma space space space straight b with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space 3 space straight j with hat on top minus 3 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 4 3 cell negative 1 end cell row 2 cell negative 3 end cell 3 end table close vertical bar space equals space straight i with hat on top space open vertical bar table row 3 cell negative 1 end cell row cell negative 3 end cell 3 end table close vertical bar space minus space straight j with hat on top space open vertical bar table row 4 cell negative 1 end cell row 2 cell negative 3 end cell end table close vertical bar plus space straight k with hat on top space open vertical bar table row 4 3 row 2 cell negative 3 end cell end table close vertical bar space space space space space space space space equals space left parenthesis negative 9 minus 3 right parenthesis space straight i with hat on top space minus space left parenthesis negative 12 plus 2 right parenthesis space straight j with hat on top space plus space left parenthesis negative 12 minus 6 right parenthesis space straight k with hat on top therefore space space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space minus 12 space straight i with hat on top space plus space 10 space straight j with hat on top space minus space 18 space straight k with hat on top space space space open vertical bar straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top close vertical bar space equals space square root of left parenthesis negative 12 right parenthesis squared plus left parenthesis 10 right parenthesis squared plus left parenthesis negative 18 right parenthesis squared end root space equals space square root of 144 plus 100 plus 324 end root space space space space space space space space space space space space space space space space space space equals space square root of 568 space equals space square root of 4 space cross times space 142 end root space equals space 2 space square root of 142
Unit vector perp to vectors straight a with rightwards arrow on top space and space straight b with rightwards arrow on top space equals space plus-or-minus space 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 space straight b with rightwards arrow on top close vertical bar end fraction
                                   equals space plus-or-minus space fraction numerator 1 over denominator 2 space square root of 142 end fraction left parenthesis negative 12 space straight i space plus space 10 space straight j with hat on top space minus space 18 space straight k with hat on top right parenthesis space equals space plus-or-minus space fraction numerator 1 over denominator square root of 142 end fraction left parenthesis negative 6 space straight i with hat on top space plus space 5 space straight j with hat on top space minus space 9 space straight k with hat on top right parenthesis

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