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Vector Algebra

Question
CBSEENMA12034119
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Show that 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 comma space space 1 over 7 left parenthesis 3 straight i with hat on top minus 6 straight j with hat on top plus 2 straight k with hat on top right parenthesis comma space space 1 over 7 left parenthesis 6 straight i with hat on top plus 2 straight j with hat on top minus 3 straight k with hat on top right parenthesis are mutually perpendicular unit vectors.

Solution

Let 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 space 2 over 7 straight i with hat on top space plus space 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 minus space 6 straight j with hat on top space plus space 2 straight k with hat on top right parenthesis space equals 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 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 plus 2 over 7 straight j with hat on top minus 3 over 7 straight k with hat on top
therefore space straight a space equals space square root of open parentheses 2 over 7 close parentheses squared plus space 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 space square root of 49 over 49 end root space equals space 1
     straight b space 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 square root of 49 over 49 end root space equals space 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 space 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 space 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 space 6 over 49 minus 18 over 49 plus 12 over 49 space equals space 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 plus open parentheses 3 over 7 close parentheses open parentheses 2 over 7 close parentheses plus open parentheses 6 over 7 close parentheses space open parentheses fraction numerator negative 3 over denominator 7 end fraction close parentheses space equals space 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 space equals 18 over 49 minus 12 over 49 minus 6 over 49 equals 0
therefore space 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 mutually space orthogonal space vectors.

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