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Vector Algebra

Question
CBSEENMA12034006
Wired Faculty App

If the vertices of a triangle are the points with position vectors 
straight a subscript 1 straight i with hat on top space plus space straight a subscript 2 straight j with hat on top space plus space straight a subscript 3 straight k with hat on top comma space space space space straight b subscript 1 straight i with hat on top space plus space straight b subscript 2 straight j with hat on top space plus space straight b subscript 3 straight k with hat on top comma space space straight c subscript 1 straight i with hat on top space plus space straight c subscript 2 straight j with hat on top space plus space straight c subscript 3 straight k with hat on top comma space space what are the vectors determined by its sides? Find the length of these vectors. 

Solution

Let A, B, C be vertices of given triangle with position vectors straight a with rightwards arrow on top space equals straight a subscript 1 space straight i with hat on top space plus space straight a subscript 2 straight j with hat on top space plus space straight a subscript 3 straight k with hat on top,
straight b with rightwards arrow on top space equals space straight b subscript 1 space straight i with hat on top space plus space straight b subscript 2 space straight j with hat on top space plus space straight b subscript 3 straight k with hat on top comma space space straight c with rightwards arrow on top space equals space straight c subscript 1 straight i with hat on top space plus space straight c subscript 2 straight j with hat on top space plus space straight c subscript 3 straight k with hat on top space respectively. space
 Now comma space space space space AB with rightwards arrow on top space equals space straight P. straight V. space of space straight B space minus space straight P. straight V. space of space straight A space equals space straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top space space space space space space equals space left parenthesis straight b subscript 1 space minus space straight a subscript 1 right parenthesis straight i with hat on top space plus space left parenthesis straight b subscript 2 minus straight a subscript 2 right parenthesis space straight j with hat on top space plus space left parenthesis straight b subscript 3 minus straight a subscript 3 right parenthesis space straight k with rightwards arrow on top BC with rightwards arrow on top space equals space straight P. straight V. space of space straight C space minus space straight P. straight V. space of space straight B space equals space straight c with rightwards arrow on top space minus space straight b with rightwards arrow on top space space space space space space space equals space left parenthesis straight c subscript 1 minus straight b subscript 1 right parenthesis space space straight i with hat on top space plus space left parenthesis straight c subscript 2 minus straight b subscript 2 right parenthesis space straight j with hat on top space plus space left parenthesis straight c subscript 3 minus straight b subscript 3 right parenthesis space straight k with hat on top CA with rightwards arrow on top space equals space straight P. straight V. space of space straight A space minus space straight P. straight V. space of space straight C space equals space straight a with rightwards arrow on top space minus space straight c with rightwards arrow on top space space space space space space equals space left parenthesis straight a subscript 1 space minus space straight c subscript 1 right parenthesis space straight i with hat on top space plus space left parenthesis straight a subscript 2 minus straight c subscript 2 right parenthesis space straight j with hat on top space plus space left parenthesis straight a subscript 3 minus straight c subscript 3 right parenthesis space straight k with hat on top
therefore space space space AB space equals space square root of left parenthesis straight b subscript 1 minus straight a subscript 1 right parenthesis squared plus left parenthesis straight b subscript 2 minus straight a subscript 2 right parenthesis squared plus left parenthesis straight b subscript 3 minus straight a subscript 3 right parenthesis squared end root space space space space space space BC space equals space square root of left parenthesis straight c subscript 1 minus straight b subscript 1 right parenthesis squared space plus space left parenthesis straight c subscript 2 minus straight b subscript 2 right parenthesis squared plus left parenthesis straight c subscript 3 minus straight b subscript 3 right parenthesis squared end root space space space space space CA space equals space square root of left parenthesis straight a subscript 1 minus straight c subscript 1 right parenthesis squared plus left parenthesis straight a subscript 2 minus straight c subscript 2 right parenthesis squared plus left parenthesis straight a subscript 3 minus straight c subscript 3 right parenthesis squared end root

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