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Vector Algebra

Question
CBSEENMA12034195
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Find a vector of magnitude 19 and which is perpendicular to both the vectors 4 space straight i with hat on top space minus space straight j with hat on top space plus space 8 space straight k with hat on top comma space space space straight b with rightwards arrow on top space equals space minus straight j with hat on top space plus 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 minus space straight j with hat on top space plus space 8 space straight k with hat on top comma space space space straight b with rightwards arrow on top space equals space minus straight j with hat on top space plus space straight k with hat on top
straight a with rightwards arrow on top cross times 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 cell negative 1 end cell 8 row 0 cell negative 1 end cell 1 end table close vertical bar space equals straight i with hat on top space open vertical bar table row cell negative 1 end cell 8 row cell negative 1 end cell 1 end table close vertical bar space minus straight j with hat on top space open vertical bar table row 4 8 row 0 1 end table close vertical bar plus space straight k with hat on top open vertical bar table row 4 cell negative 1 end cell row 0 cell negative 1 end cell end table close vertical bar
            equals left parenthesis negative 1 plus 8 right parenthesis space straight i with hat on top space minus left parenthesis 4 minus 0 right parenthesis space straight j with hat on top plus left parenthesis negative 4 plus 0 right parenthesis space straight k with hat on top space equals space 7 space straight i with hat on top space minus space 4 space straight j with hat on top space minus space 4 space straight k with hat on top
which is vector perpendicular to straight a with rightwards arrow on top space and space straight b with rightwards arrow on top .
              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 7 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared end root space equals space square root of 49 plus 16 plus 16 end root space equals square root of 81 space equals space 9
therefore a unit vector perpendicular to straight a with rightwards arrow on top space and space straight b with rightwards arrow on top is
                     equals 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 cross times space straight b with rightwards arrow on top close vertical bar end fraction space equals space 1 over 9 left parenthesis 7 straight i with hat on top space minus 4 space straight j with hat on top space minus space 4 straight k with hat on top right parenthesis
therefore a vector of magnitude of 19 and perpendicular to straight a with rightwards arrow on top space and space straight b with rightwards arrow on top is 
                             equals 19 space open parentheses 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 close parentheses space equals 19 over 9 left parenthesis 7 space straight i with hat on top space minus space 4 space straight j with hat on top space minus space 4 space straight k with hat on top right parenthesis

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