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Vector Algebra

Question
CBSEENMA12034193
Wired Faculty App

Find a vector of magnitude 9 and which is perpendicular to both the vectors
straight a with rightwards arrow on top space equals space 4 straight i with hat on top space minus space straight j with hat on top space plus space 3 space straight k with hat on top comma space space space straight b with rightwards arrow on top space equals space minus 2 space straight i with hat on top space space plus space straight j with hat on top space minus space 2 space straight k with hat on top
 

Solution

Here 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 3 space straight k with hat on top comma space space space space straight b with rightwards arrow on top space equals space minus 2 space straight i with hat on top space plus space straight j with hat on top space minus space 2 space straight k with hat on top
therefore space 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 3 row cell negative 2 end cell cell negative 1 end cell cell negative 2 end cell end table close vertical bar space equals space straight i with hat on top space open vertical bar table row cell negative 1 end cell 3 row 1 cell negative 2 end cell end table close vertical bar space minus straight j with hat on top space open vertical bar table row 4 3 row cell negative 2 end cell cell negative 2 end cell end table close vertical bar plus space straight k with hat on top space open vertical bar table row 4 cell negative 1 end cell row 2 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 equals space left parenthesis 2 minus 3 right parenthesis space straight i with hat on top space minus space left parenthesis negative 8 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 minus straight i with hat on top space plus space 2 space straight j with hat on top space plus space 2 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 1 plus 4 plus 4 end root space equals space square root of 9 space equals space 3
therefore space space space 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 cross times 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 1 third left parenthesis negative straight i with hat on top plus 2 space straight j with hat on top space plus space space 2 space straight k with hat on top right parenthesis
           therefore   a vector of magnitude 9 and perp. to straight a with rightwards arrow on top space and space straight b with rightwards arrow on top  is
                                      equals space 9 space open parentheses 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 close parentheses space equals 9 over 3 left parenthesis negative straight i with hat on top space plus space 2 space straight j with hat on top space plus space 2 space straight k with hat on top right parenthesis space equals space 3 space left parenthesis negative straight i with hat on top space plus space 2 straight j with hat on top space plus 2 straight k with hat on top right parenthesis
Required vector can be  = negative 3 left parenthesis negative straight i with hat on top space plus 2 space straight j with hat on top space plus space 2 space straight k with hat on top right parenthesis.

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