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Vector Algebra

Question
CBSEENMA12034130
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If 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 with rightwards arrow on top comma show that the angle straight theta between the vectors straight b with rightwards arrow on top space and space straight c with rightwards arrow on top is given by
cos space straight theta space equals space fraction numerator straight a squared minus straight b squared minus straight c squared over denominator 2 space open vertical bar straight b with rightwards arrow on top close vertical bar space open vertical bar straight c with rightwards arrow on top close vertical bar end fraction

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 space equals 0 with rightwards arrow on top
therefore space space space straight a with rightwards arrow on top space space equals space minus left parenthesis straight b with rightwards arrow on top plus straight c with rightwards arrow on top right parenthesis                                                 ...(1)
therefore space space space straight a squared space equals space straight a with rightwards arrow on top. space straight a with rightwards arrow on top space equals space open curly brackets negative left parenthesis straight b with rightwards arrow on top plus straight c with rightwards arrow on top right parenthesis close curly brackets space. space left curly bracket negative left parenthesis straight b with rightwards arrow on top plus straight c with rightwards arrow on top right parenthesis right curly bracket space space space space space space space space open square brackets because space of space left parenthesis 1 right parenthesis close square brackets
             equals space left parenthesis straight b with rightwards arrow on top plus straight c with rightwards arrow on top right parenthesis. 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. 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 b with rightwards arrow on top space plus straight c with rightwards arrow on top. straight c with rightwards arrow on top equals space straight b with rightwards arrow on top. straight b with rightwards arrow on top space plus straight b with rightwards arrow on top. space straight c 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 c with rightwards arrow on top space equals space straight b with rightwards arrow on top. straight b with rightwards arrow on top plus 2 left parenthesis straight b with rightwards arrow on top minus straight c with rightwards arrow on top right parenthesis plus straight c with rightwards arrow on top. straight c with rightwards arrow on top
therefore space space space space space straight a squared space equals space straight b squared plus 2 space open vertical bar straight b with rightwards arrow on top close vertical bar space open vertical bar straight c with rightwards arrow on top close vertical bar space cos space straight theta space plus space straight c squared
space therefore space space 2 space open vertical bar straight b with rightwards arrow on top close vertical bar space open vertical bar straight c with rightwards arrow on top close vertical bar space cos space straight theta space equals straight a squared minus straight b squared minus straight c squared space therefore space space space space cos space straight theta space equals space fraction numerator straight a squared minus straight b squared minus straight c squared over denominator 2 space open vertical bar straight b with rightwards arrow on top close vertical bar space open vertical bar straight c with rightwards arrow on top close vertical bar end fraction

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