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Vector Algebra

Question
CBSEENMA12034117
Wired Faculty App

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

Solution

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

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