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Vector Algebra

Question
CBSEENMA12034185
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Find a unit vector perpendicular to each of the vector straight a with rightwards arrow on top plus straight b with rightwards arrow on top space and space straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top comma where straight a with rightwards arrow on top space equals space 3 straight i with hat on top space plus space 2 straight j with hat on top space plus space 2 straight k with hat on top and straight b with rightwards arrow on top space equals space straight i with hat on top space plus space 2 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 3 straight i with hat on top plus 2 straight j with hat on top plus 2 straight k with hat on top comma space space space space space space straight b with rightwards arrow on top space equals space straight i with hat on top space plus space 2 straight j with hat on top space minus space 2 straight k with hat on top
therefore space space space space space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space 4 straight i with hat on top space plus space 4 straight j with hat on top comma space space space space straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top space equals 2 space straight i with hat on top space plus space 4 space straight k with hat on top
left parenthesis straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top right parenthesis space cross times space space left parenthesis straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top right parenthesis 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 4 0 row 2 0 4 end table close vertical bar space space equals space straight i with hat on top space open vertical bar table row 4 0 row 0 4 end table close vertical bar space minus space straight j with hat on top space open vertical bar table row 4 0 row 2 4 end table close vertical bar space plus space straight k with hat on top space open vertical bar table row 4 4 row 2 0 end table close vertical bar space space space space space space space space space space space equals space left parenthesis 16 space minus space 0 right parenthesis space straight i with hat on top space minus space left parenthesis 16 minus 0 right parenthesis space straight j with hat on top space plus space left parenthesis 0 minus 8 right parenthesis space straight k with hat on top space equals space 16 space straight i with hat on top space minus space 16 space straight j with hat on top space minus space 8 space straight k with hat on top
which is a vector perpendicular to straight a with rightwards arrow on top as well as straight b with rightwards arrow on top.
left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis space cross times space left parenthesis straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top right parenthesis 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 4 0 row 2 0 4 end table close vertical bar space equals space straight i with hat on top space open vertical bar table row 4 0 row 0 4 end table close vertical bar space minus space straight j with hat on top space open vertical bar table row 4 0 row 2 4 end table close vertical bar space plus space straight k with hat on top space open vertical bar table row 4 4 row 2 0 end table close vertical bar space space space space space space equals space left parenthesis 16 minus 0 right parenthesis space straight i with hat on top space minus space left parenthesis 16 minus 0 right parenthesis space straight j with hat on top space plus space left parenthesis 0 minus 8 right parenthesis space straight k with hat on top space equals space 16 space straight i with hat on top space minus space 16 space straight j with hat on top space minus 8 space straight k with hat on top
which is a vector perpendicular to straight a with rightwards arrow on top space as space well space as space straight b with rightwards arrow on top.
open vertical bar left parenthesis straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top right parenthesis space cross times space left parenthesis straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top right parenthesis close vertical bar space equals space square root of left parenthesis 16 right parenthesis squared plus left parenthesis negative 16 right parenthesis squared plus left parenthesis negative 8 right parenthesis squared end root space equals square root of 256 plus 256 plus 64 end root 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 equals space square root of 576 space equals space 24
therefore a unit vector perpendicular to left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis space and space left parenthesis straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top right parenthesis is 
                  equals space fraction numerator left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis space cross times space left parenthesis straight a with rightwards arrow on top minus straight b with rightwards arrow on top right parenthesis over denominator open vertical bar left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis space cross times space space left parenthesis straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top right parenthesis close vertical bar end fraction space equals space 1 over 24 left parenthesis 16 space straight i with hat on top space minus space 16 space straight j with hat on top space minus space 8 space straight k with hat on top right parenthesis space equals space 1 third left parenthesis 2 straight i with hat on top minus 2 straight j with hat on top minus straight k with hat on top right parenthesis

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