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Vector Algebra

Question
CBSEENMA12034264
Wired Faculty App

Give space space space straight a with rightwards arrow on top space equals space 1 over 7 left parenthesis 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 6 space straight k with hat on top right parenthesis comma space space straight b with rightwards arrow on top space equals space 1 over 7 left parenthesis 3 space straight i with hat on top space minus space 6 space straight j with hat on top space plus space 2 space straight k with hat on top right parenthesis comma space space space space space space space space space space space space space space straight c with rightwards arrow on top space equals space 1 over 7 left parenthesis 6 space straight i with hat on top space plus space 2 space straight j with hat on top space minus space 3 space straight k with hat on top right parenthesis comma
straight i with hat on top comma space straight j with hat on top comma space straight k with hat on top being a right-handed orthogonal system of unit vectors in space. Show that straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top in another such system.

Solution

        straight a with rightwards arrow on top space equals space 1 over 7 left parenthesis 2 straight i with hat on top space plus space 3 straight j with hat on top space plus space 6 straight k with hat on top right parenthesis space equals 2 over 7 straight i with hat on top plus 3 over 7 straight j with hat on top space plus space 6 over 7 straight k with hat on top straight b with rightwards arrow on top space equals space 1 over 7 left parenthesis 3 straight i with hat on top space minus space 6 straight j with hat on top space plus space 2 straight k with hat on top right parenthesis space equals space 3 over 7 straight i with hat on top space minus space 6 over 7 straight j with hat on top space plus space 2 over 7 straight k with hat on top straight c with rightwards arrow on top space equals space 1 over 7 left parenthesis 6 straight i with hat on top space plus space 2 straight j with hat on top space minus space 3 straight k with hat on top right parenthesis space equals space 6 over 7 straight i with hat on top space plus space 2 over 7 straight j with hat on top space minus space 3 over 7 straight k with hat on top
therefore space space space space straight a space equals space square root of open parentheses 2 over 7 close parentheses squared plus open parentheses 3 over 7 close parentheses squared plus open parentheses 6 over 7 close parentheses squared end root space equals space square root of 4 over 49 plus 9 over 49 plus 36 over 49 end root space equals square root of 49 over 49 end root space equals 1
          straight b equals space square root of open parentheses 3 over 7 close parentheses squared plus open parentheses fraction numerator negative 6 over denominator 7 end fraction close parentheses squared plus open parentheses 2 over 7 close parentheses squared end root space equals space square root of 9 over 49 plus 36 over 49 plus 4 over 49 end root space equals space square root of 49 over 49 end root space equals 1 straight c space equals space square root of open parentheses 6 over 7 close parentheses squared plus open parentheses 2 over 7 close parentheses squared plus open parentheses fraction numerator negative 3 over denominator 7 end fraction close parentheses squared end root space equals square root of 36 over 49 plus 4 over 49 plus 9 over 49 end root space equals square root of 49 over 49 end root space equals space 1
therefore space space straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top space are space unit space vectors.
                straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space open parentheses 2 over 7 close parentheses space open parentheses 3 over 7 close parentheses plus space open parentheses 3 over 7 close parentheses space open parentheses fraction numerator negative 6 over denominator 7 end fraction close parentheses space plus space open parentheses 6 over 7 close parentheses space open parentheses 2 over 7 close parentheses space equals 6 over 49 minus 18 over 49 plus 12 over 49 equals 0 straight a with rightwards arrow on top. space straight c with rightwards arrow on top space equals space open parentheses 2 over 7 close parentheses space open parentheses 6 over 7 close parentheses space plus space open parentheses 3 over 7 close parentheses space open parentheses 2 over 7 close parentheses space plus open parentheses 6 over 7 close parentheses space open parentheses fraction numerator negative 3 over denominator 7 end fraction close parentheses space equals 12 over 49 plus 6 over 49 minus 18 over 49 space equals 0 straight b with rightwards arrow on top. straight c with rightwards arrow on top space equals space open parentheses 3 over 7 close parentheses space open parentheses 6 over 7 close parentheses plus open parentheses fraction numerator negative 6 over denominator 7 end fraction close parentheses space open parentheses 2 over 7 close parentheses plus open parentheses 2 over 7 close parentheses space open parentheses fraction numerator negative 3 over denominator 7 end fraction close parentheses space equals space 18 over 49 minus 12 over 49 minus 6 over 49 equals space 0
therefore space space straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top are mutually orthogonal vectors.
Now,  
   straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space 1 over 49 open vertical bar table row cell straight i with hat on top end cell cell space straight j with hat on top end cell cell straight k with hat on top end cell row 2 cell space space 3 end cell 6 row 3 cell negative 6 end cell 2 end table close vertical bar space equals space 1 over 49 open square brackets left parenthesis 6 plus 36 right parenthesis space straight i with hat on top space minus space left parenthesis 4 minus 18 right parenthesis space straight j with hat on top space plus space left parenthesis negative 12 minus 9 right parenthesis straight k with hat on top close square brackets space space space space space space space space space space space space space equals space 1 over 49 left square bracket 42 straight i with hat on top space plus space 14 straight j with hat on top space minus space 21 straight k with hat on top right square bracket space equals space 1 over 7 left parenthesis 6 straight i with hat on top space plus 2 straight j with hat on top space minus 3 straight k with hat on top right parenthesis space equals space straight c with rightwards arrow on top straight b with rightwards arrow on top space cross times space straight c with rightwards arrow on top space equals space 1 over 49 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 3 cell negative 6 end cell 2 row 6 2 cell negative 3 end cell end table close vertical bar space equals 1 over 49 open square brackets left parenthesis 18 minus 4 right parenthesis straight i with hat on top space minus space left parenthesis negative 9 minus 12 right parenthesis space straight j with hat on top space plus space left parenthesis 6 plus 36 right parenthesis space straight k with hat on top close square brackets space space space space space space space space space space space space space space equals space 1 over 49 left square bracket 14 space straight i with hat on top space plus 21 space straight j with hat on top space plus space 42 space straight k with hat on top right square bracket space equals 1 over 7 left parenthesis 2 straight i with hat on top space plus space 3 straight j with hat on top space plus 6 space straight k with hat on top right parenthesis space equals space straight a with rightwards arrow on top straight c with rightwards arrow on top cross times space straight a with rightwards arrow on top space equals 1 over 49 open vertical bar table row cell straight i with hat on top end cell cell space straight j with hat on top end cell cell straight k with hat on top end cell row 6 cell space space 2 end cell cell negative 3 end cell row 2 cell space space 3 end cell cell space space 6 end cell end table close vertical bar space equals space 1 over 49 open square brackets left parenthesis 12 plus 9 right parenthesis space straight i with hat on top space minus space left parenthesis 36 plus 6 right parenthesis space straight j with hat on top space plus space left parenthesis 18 minus 4 right parenthesis straight k with hat on top close square brackets space space space space space space space space space space space space space equals space 1 over 49 left square bracket 21 space straight i with hat on top space minus space 42 space straight j with hat on top space plus space 14 space straight k with hat on top right square bracket space equals space 1 over 7 left parenthesis 3 space straight i with hat on top minus 6 space straight j with hat on top space plus 2 space straight k with hat on top right parenthesis space equals straight b with rightwards arrow on top
therefore space space straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top from a right handed orthogonal system of unit vectors. 


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