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Vector Algebra

Question
CBSEENMA12034009
Wired Faculty App

Prove that the points straight i with hat on top space minus space straight j with hat on top comma space space 4 straight i with hat on top space minus 3 space straight j with hat on top space plus space straight k with hat on top space and 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 space straight k with hat on top are 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 straight i with hat on top space minus space straight j with hat on top comma space space space straight b 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 straight c with rightwards arrow on top space equals space 2 space straight i with hat on top space space minus space 4 space straight j with hat on top space plus space 5 space straight k with hat on top 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 space space equals space left parenthesis 4 space straight i with hat on top space minus space 3 space straight j with hat on top space plus space space straight k with hat on top right parenthesis space minus space left parenthesis straight i with hat on top space minus space space straight j with hat on top right parenthesis space equals space 3 space straight i with hat on top space minus space 2 space straight j with hat on top space plus 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 space space space space space space space equals space left parenthesis 2 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 right parenthesis space minus space left parenthesis 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 right parenthesis space equals space minus space 2 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 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 space space space space space equals left parenthesis straight i with hat on top space minus space straight j with hat on top right parenthesis space minus space left parenthesis 2 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 right parenthesis 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 therefore space 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 ABC space is space straight a space triangle.
                    AB space equals space square root of 9 plus 4 plus 1 end root space equals space square root of 14 BC space equals space square root of 4 plus 1 plus 16 end root space equals space square root of 21 CA space equals space square root of 1 plus 9 plus 25 end root space equals square root of 35
Now,  AB squared plus BC squared space equals space CA squared
therefore space space space increment ABC space is space right space angled space at space straight B.

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