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Vector Algebra

Question
CBSEENMA12034135
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If 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 unit vectors such that straight a with rightwards arrow on top plus straight b with rightwards arrow on top plus straight c with rightwards arrow on top space equals space 0, then find the value of straight a with rightwards arrow on top. straight b with rightwards arrow on top plus straight b with rightwards arrow on top. straight c with rightwards arrow on top plus straight c with rightwards arrow on top. straight a with rightwards arrow on top.

Solution

Here
               straight a with rightwards arrow on top plus straight b with rightwards arrow on top plus straight c with rightwards arrow on top space equals 0 with rightwards arrow on top
therefore space space space straight a with rightwards arrow on top plus straight b with rightwards arrow on top space equals negative straight c with rightwards arrow on top                                 ...(1)
therefore space space space straight a with rightwards arrow on top. space left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis space equals space minus straight a with rightwards arrow on top. straight c with rightwards arrow on top rightwards double arrow space space space straight a with rightwards arrow on top. space 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 equals space minus straight a with rightwards arrow on top. space straight c with rightwards arrow on top rightwards double arrow space open vertical bar straight a with rightwards arrow on top close vertical bar squared plus straight a with rightwards arrow on top. straight b with rightwards arrow on top plus straight a with rightwards arrow on top. straight c with rightwards arrow on top space equals 0 therefore space space space 1 plus straight a with rightwards arrow on top. straight b with rightwards arrow on top plus straight a with rightwards arrow on top. straight c with rightwards arrow on top space equals 0 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 straight a with rightwards arrow on top space is space straight a space unit space vector close square brackets From space left parenthesis 1 right parenthesis comma space space straight b with rightwards arrow on top. space left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis space equals negative straight b with rightwards arrow on top. space straight c with rightwards arrow on top space space space rightwards double arrow space space space straight b with rightwards arrow on top. space stack straight a space with rightwards arrow on top space plus space straight b with rightwards arrow on top. space straight b with rightwards arrow on top space equals space minus straight b with rightwards arrow on top. space straight c with rightwards arrow on top rightwards double arrow space space space straight b with rightwards arrow on top. space straight a with rightwards arrow on top plus space open vertical bar straight b with rightwards arrow on top close vertical bar squared space plus space straight b with rightwards arrow on top. space straight c with rightwards arrow on top space equals space 0 rightwards double arrow space space straight b with rightwards arrow on top. space straight a with rightwards arrow on top space plus 1 plus space straight b with rightwards arrow on top space. space straight c 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 space space open square brackets because space straight b with rightwards arrow on top space is space straight a space unit space vector close square brackets therefore space space space 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 c with rightwards arrow on top space equals space minus 1 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 space space space space space space space space space... left parenthesis 3 right parenthesis
Again from (1), straight c with rightwards arrow on top. space left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis space equals space minus straight c with rightwards arrow on top. space straight c with rightwards arrow on top
therefore space space 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 minus open vertical bar straight c with rightwards arrow on top close vertical bar squared space space space space rightwards double arrow space space space 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 plus space open vertical bar straight c with rightwards arrow on top close vertical bar squared space equals space 0 rightwards double arrow space space straight c with rightwards arrow on top. space straight a with rightwards arrow on top space plus straight c with rightwards arrow on top. space straight b with rightwards arrow on top space plus 1 space space equals space 0 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 space space space space space space space space space space space space open square brackets because space straight c with rightwards arrow on top space is space straight a space unit space vector close square brackets rightwards double arrow space 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 minus 1
Adding (2), (3) and (4), we get
                 2 left parenthesis straight a 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. space straight a with rightwards arrow on top right parenthesis space equals space minus 3 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 space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space straight b with rightwards arrow on top. space straight a with rightwards arrow on top space etc. close square brackets therefore space straight a 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. space straight a with rightwards arrow on top space equals space minus 3 over 2

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