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Vector Algebra

Question
CBSEENMA12034174
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Find the unit vector perpendicular to vectors 4 space straight i with hat on top space plus space straight j with hat on top space space minus space 3 space straight k with hat on top comma space space space space 2 space straight i with hat on top space plus space straight j with hat on top space minus space 3 space straight k with hat on top.

Solution

Let straight a with rightwards arrow on top space equals space 4 space straight i with hat on top space plus space straight j with hat on top space minus space 3 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 plus space straight j with hat on top space minus space 3 space straight k with hat on top
 therefore space space space 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 1 cell negative 3 end cell row 2 1 cell negative 3 end cell end table close vertical bar space equals space straight i with hat on top space open vertical bar table row 1 cell negative 3 end cell row 1 cell negative 3 end cell end table close vertical bar space minus straight j with hat on top space open vertical bar table row 4 cell negative 3 end cell row 2 cell negative 3 end cell end table close vertical bar space plus space straight k with hat on top space open vertical bar table row 4 1 row 2 1 end table close vertical bar space space space space space space space space space space space space space space space space space space equals left parenthesis negative 3 plus 3 right parenthesis space straight i with hat on top space minus space left parenthesis negative 12 plus 6 right parenthesis space straight j with hat on top space plus space left parenthesis 4 minus 2 right parenthesis space straight k with hat on top space equals space 6 space straight j with hat on top space plus space 2 space straight k with hat on top therefore space space space space space space open vertical bar straight a with rightwards arrow on top cross times straight b with rightwards arrow on top close vertical bar space equals space square root of 0 plus 36 plus 4 end root space equals space square root of 40 Unit space vector space perpendicular space to space straight a with rightwards arrow on top space and space space straight b with rightwards arrow on top space equals space fraction numerator straight a with rightwards arrow on top cross times 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 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 plus-or-minus fraction numerator 1 over denominator square root of 40 end fraction left parenthesis 6 space straight j with hat on top space plus space 2 space straight k with hat on top right parenthesis space equals space plus-or-minus fraction numerator 1 over denominator square root of 10 end fraction left parenthesis 3 space straight j with hat on top space plus space straight k with hat on top right parenthesis 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

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