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Vector Algebra

Question
CBSEENMA12034008

Show that the points A, B and C with position vectors, respectively form the vertices of a right angled triangle, stack space straight a 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 comma space space straight b with rightwards arrow on top space equals space 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 space and space straight c with rightwards arrow on top space equals space straight i with hat on top space minus space 3 space straight j with hat on top space space minus space 5 space straight k with hat on top comma space respectively form the vertices of a right angled triangle. 

Solution
Let A, B, C, be the points whose position vectors are
                      straight a 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
straight b with rightwards arrow on top space equals space 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
straight c with rightwards arrow on top space equals space 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 space respectively
     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 minus straight a with rightwards arrow on top space equals space minus straight i with hat on top plus 3 space straight j with hat on top plus 5 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 space minus space straight b with rightwards arrow on top space equals space minus straight i with hat on top space minus space 2 space straight j with hat on top space minus space 6 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 equals space 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
therefore space space AB with rightwards arrow on top space plus space BC with rightwards arrow on top space plus space CA with rightwards arrow on top space equals space 0 with rightwards arrow on top
therefore space space space space space ABC space is space straight a space triangle.
space space space space space space space space AB space equals space square root of left parenthesis negative 1 right parenthesis squared plus left parenthesis 3 right parenthesis squared plus left parenthesis 5 right parenthesis squared end root space equals space square root of 1 plus 9 plus 25 end root space equals space square root of 35
space space space space space space space space BC space equals space square root of left parenthesis negative 1 right parenthesis squared plus left parenthesis negative 2 right parenthesis squared plus left parenthesis negative 6 right parenthesis squared end root space equals space square root of 1 plus 4 plus 36 end root space equals space square root of 41
space space space space space space space space space CA space equals space square root of left parenthesis 2 right parenthesis squared plus left parenthesis negative 1 right parenthesis squared plus left parenthesis 1 right parenthesis squared end root space equals space square root of 4 plus 1 plus 1 end root space equals space square root of 6
Now,  CA squared plus AB squared space equals space 6 plus 35 space equals 41 space equals space BC squared
therefore space space space space increment ABC space is space right space angled space at space straight A.