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Vector Algebra

Question
CBSEENMA12034115
Wired Faculty App

Show that the vectors 2 straight i with hat on top minus straight j with hat on top plus straight k with hat on top comma space straight b with rightwards arrow on top space equals space straight i with hat on top space minus space 3 straight j with hat on top space minus space 5 straight k with hat on top comma space straight c with rightwards arrow on top space equals space 3 straight i with hat on top minus 4 straight j with hat on top minus 4 straight k with hat on top form the vertices of a right angled triangle. 

Solution

Let 
straight a with rightwards arrow on top space equals 2 straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top comma space space space straight b with rightwards arrow on top space equals straight i with hat on top minus 3 space straight j with hat on top space minus space 5 space straight k with hat on top comma space space space straight c with rightwards arrow on top space equals space 3 space straight i with hat on top space minus space 4 space straight j with hat on top space minus space 4 space straight k with hat on top
be the position vectors of A. B, C respectively.
therefore 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 straight b with rightwards arrow on top minus straight a with rightwards arrow on top
                              equals left parenthesis straight i with hat on top space space minus space 3 space straight j with hat on top space minus space space 5 straight k with hat on top right parenthesis space minus space left parenthesis 2 straight i with hat on top minus straight j with hat on top plus straight k with hat on top right parenthesis space equals space minus straight i with hat on top minus 2 space straight j with hat on top space minus space 6 space straight k with hat 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 minus straight b with rightwards arrow on top space space space space space space space equals space left parenthesis 3 space straight i with hat on top space minus space 4 space straight j with hat on top space minus space space 4 space straight k with hat on top right parenthesis space minus space left parenthesis straight i with hat on top space minus space 3 space straight j with hat on top space minus space 5 space straight k with hat on top right parenthesis space equals space 2 straight i with hat on top space minus space straight j with hat on top space plus 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 minus straight c with rightwards arrow on top space space space space space space space equals space left parenthesis 2 space straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top right parenthesis space minus space left parenthesis 3 space straight i with hat on top space minus space 4 space straight j with hat on top space minus space 4 space straight k with hat on top right parenthesis space equals space minus straight i with hat on top space plus space 3 space straight j with hat on top space plus space 5 space straight k with hat on top therefore space space space AB with rightwards arrow on top plus BC with rightwards arrow on top space equals straight i with hat on top space space minus space 3 space straight j with hat on top space minus space 5 space straight k with hat on top space equals space minus CA with rightwards arrow on top therefore space space space AB with rightwards arrow on top plus BC with rightwards arrow on top plus CA with rightwards arrow on top space equals space 0 with rightwards arrow on top Again comma space space BC with rightwards arrow on top. space CA with rightwards arrow on top space equals space left parenthesis 2 right parenthesis thin space left parenthesis negative 1 right parenthesis space plus space left parenthesis negative 1 right parenthesis thin space left parenthesis 3 right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis 5 right parenthesis space equals negative 2 minus 3 plus 5 space equals space 0 therefore space space space space BC with rightwards arrow on top space and space CA with rightwards arrow on top space are space perpendicular therefore space space space increment ABC space is space right space angled
therefore given position vectors 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 form the vertices of a right angled triangle.

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