Question

Show that the vectors and $straight a with rightwards arrow on top space equals space 3 space straight i with hat on top space minus space 4 space straight j with hat on top space minus space 4 space straight k with hat on top comma space space space space straight b with rightwards arrow on top space equals space 2 straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top comma space space straight c with rightwards arrow on top space equals space straight i with hat on top space minus space 3 space straight j with hat on top space minus space 5 space straight k with hat on top$ form the sides of 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 straight a squared space equals space left parenthesis 3 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared space equals space 9 plus 16 plus 16 space equals space 41 straight b squared space equals space left parenthesis 2 right parenthesis squared plus left parenthesis negative 1 right parenthesis squared plus left parenthesis 1 right parenthesis squared space equals space 4 plus 1 plus 1 space equals space 6 straight c squared space equals space left parenthesis 1 right parenthesis squared plus left parenthesis negative 3 right parenthesis squared plus left parenthesis negative 5 right parenthesis squared space equals space 1 plus 9 plus 25 space equals space 35

Now, 6 + 35 = 41

therefore space space space straight b squared space plus space straight c squared equals space space straight a squared therefore space space space space 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 space form space the space sides space of space straight a space right space angled space triangle. space

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