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Vector Algebra

Question
CBSEENMA12034143
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Using vectors, prove cosine formula 
(i) cos space straight A space equals space fraction numerator straight b squared plus straight c squared minus straight a squared over denominator 2 space straight b space straight c end fraction

(ii) cos space straight B space equals fraction numerator straight c squared plus straight a squared minus straight b squared over denominator 2 ca end fraction

Solution

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

(ii)    Now straight c with rightwards arrow on top plus straight a with rightwards arrow on top space equals space minus straight b with rightwards arrow on top            rightwards double arrow space space space left parenthesis straight c with rightwards arrow on top plus straight a with rightwards arrow on top right parenthesis squared space equals space left parenthesis negative straight b with rightwards arrow on top right parenthesis squared
rightwards double arrow space space space space straight c with rightwards arrow on top squared plus straight a with rightwards arrow on top squared space plus space 2 space straight c with rightwards arrow on top. space straight a with rightwards arrow on top space equals space straight b with rightwards arrow on top squared space space space space space space space space rightwards double arrow space space straight c squared plus straight a squared plus 2 space straight c space straight a space left parenthesis 180 degree minus straight B right parenthesis space equals straight b squared rightwards double arrow space space space space straight c squared plus straight a squared minus 2 space straight c space straight a space cos space straight B space equals space straight b squared space space space space space space rightwards double arrow space space space 2 space straight c space straight a space cos space straight B space equals straight c squared plus straight a squared minus straight b squared therefore space space space cos space straight B space equals space fraction numerator straight c squared plus straight a squared minus straight b squared over denominator 2 ca end fraction.

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