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Vector Algebra

Question
CBSEENMA12034090
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If open vertical bar straight a with rightwards arrow on top close vertical bar space equals space 3 comma space space open vertical bar straight b with rightwards arrow on top close vertical bar space equals space 4 space and space open vertical bar straight c with rightwards arrow on top close vertical bar space equals 5 such that each is perpendicular to sum of the other two, find open vertical bar straight a with rightwards arrow on top plus straight b with rightwards arrow on top plus straight c with rightwards arrow on top close vertical bar.

Solution

We have
        open vertical bar straight a with rightwards arrow on top close vertical bar space equals 3 comma space space space open vertical bar straight b with rightwards arrow on top close vertical bar space equals space 4 comma space space open vertical bar straight c with rightwards arrow on top close vertical bar space equals space 5                            ...(1)
 Since straight a with rightwards arrow on top is perpendicular to straight b with rightwards arrow on top space plus space straight c with rightwards arrow on top comma space straight b with rightwards arrow on top space is space perp comma space straight c with rightwards arrow on top plus straight a with rightwards arrow on top space and space straight c with rightwards arrow on top space is space perp. space to space straight a with rightwards arrow on top plus straight b with rightwards arrow on top
therefore space space space straight a with rightwards arrow on top. space left parenthesis straight b with rightwards arrow on top plus straight c with rightwards arrow on top right parenthesis space equals space straight b with rightwards arrow on top. space space space left parenthesis straight c with rightwards arrow on top space plus space straight a with rightwards arrow on top right parenthesis space equals space straight c with rightwards arrow on top. space left parenthesis straight a with rightwards arrow on top plus space straight b with rightwards arrow on top right parenthesis space equals space 0 therefore space space space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space plus space straight a with rightwards arrow on top. space straight c with rightwards arrow on top space equals space straight b with rightwards arrow on top. space straight c with rightwards arrow on top space plus space straight b with rightwards arrow on top. space straight a with rightwards arrow on top space equals space straight c with rightwards arrow on top. space straight a with rightwards arrow on top space plus space straight c with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 0 space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis
Now,  open vertical bar straight a with rightwards arrow on top plus straight b with rightwards arrow on top plus straight c with rightwards arrow on top close vertical bar squared space equals space left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top plus straight c with rightwards arrow on top right parenthesis. space left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top plus straight c with rightwards arrow on top right parenthesis                                
        equals space straight a with rightwards arrow on top. straight a with rightwards arrow on top space plus space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space plus space straight a with rightwards arrow on top. straight c with rightwards arrow on top space plus space straight b with rightwards arrow on top. space straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top. space straight b with rightwards arrow on top space plus space straight b with rightwards arrow on top space. straight c with rightwards arrow on top space plus space straight c with rightwards arrow on top. straight a with rightwards arrow on top space plus space straight c with rightwards arrow on top. space straight b with rightwards arrow on top space plus space straight c with rightwards arrow on top. space straight c with rightwards arrow on top space equals space open vertical bar straight a with rightwards arrow on top close vertical bar squared thin space left parenthesis straight a with rightwards arrow on top. space straight b with rightwards arrow on top space plus space straight a with rightwards arrow on top. space straight c with rightwards arrow on top right parenthesis space plus space left parenthesis straight b with rightwards arrow on top. space straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top. space straight c with rightwards arrow on top right parenthesis space plus space open vertical bar straight b with rightwards arrow on top close vertical bar squared space plus left parenthesis straight c with rightwards arrow on top. space straight a with rightwards arrow on top space plus space straight c with rightwards arrow on top. space straight b with rightwards arrow on top right parenthesis space plus space open vertical bar straight c with rightwards arrow on top close vertical bar squared space equals left parenthesis 3 right parenthesis squared plus 0 plus 0 plus left parenthesis 4 right parenthesis squared plus 0 plus left parenthesis 5 right parenthesis squared 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 open square brackets because space of space left parenthesis 1 right parenthesis comma space left parenthesis 2 right parenthesis close square brackets equals space 9 plus 16 plus 25 space equals space 50
therefore space space space space open vertical bar straight a with rightwards arrow on top plus straight b with rightwards arrow on top plus straight c with rightwards arrow on top close vertical bar space equals space square root of 50 space equals space 5 square root of 2

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