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Vector Algebra

Question
CBSEENMA12034142
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Using vectors, prove Cosine formula 
cos space straight C space equals space fraction numerator straight a squared plus straight b squared minus straight c squared over denominator 2 ab end fraction

Solution

Let BC with rightwards arrow on top space equals space straight a with rightwards arrow on top,  CA with rightwards arrow on top space equals straight b with rightwards arrow on top comma space AB with rightwards arrow on top space equals straight c with rightwards arrow on top
therefore space space space BC space equals space straight a comma space space space CA space equals space straight b comma space space space AB space equals space straight c
Now, straight a with rightwards arrow on top plus straight b with rightwards arrow on top equals negative straight c with rightwards arrow on top
rightwards double arrow space space space left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis squared space equals space left parenthesis negative straight c with rightwards arrow on top right parenthesis squared rightwards double arrow space space space space straight a with rightwards arrow on top squared plus straight b with rightwards arrow on top squared space plus space 2 space straight a with rightwards arrow on top space straight b with rightwards arrow on top space equals space straight c with rightwards arrow on top squared rightwards double arrow space space straight a squared plus straight b squared plus 2 space ab space cos left parenthesis 180 degree space minus space straight C right parenthesis space equals space straight c squared rightwards double arrow space space space straight a squared plus straight b squared minus 2 space ab space cos space straight C space equals space straight c squared rightwards double arrow space space straight a squared plus straight b squared minus 2 space ab space cos space straight C space equals straight c squared space space space space space space space space space space rightwards double arrow space space space space 2 space ab space cos space straight C space equals space straight a squared plus straight b squared minus straight c squared rightwards double arrow space space cos space straight C space equals space fraction numerator straight a squared plus straight b squared minus straight c squared over denominator 2 space straight a space straight b end fraction

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