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Vector Algebra

Question
CBSEENMA12034114
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Show that the vectors straight a 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 comma space straight b with rightwards arrow on top space equals 2 space straight i with hat on top space minus straight j with hat on top plus straight k with hat on top comma space and space straight c with rightwards arrow on top space equals straight i with hat on top plus 2 space straight j with hat on top plus 6 space straight k with hat on top form a right angled triangle. 

Solution

Here,    
        straight a 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 comma space space space straight b with rightwards arrow on top space equals 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 comma space straight c with rightwards arrow on top space equals straight i with hat on top space plus space 2 space straight j with hat on top space plus space 6 space straight k with hat on top
Now, straight c with rightwards arrow on top plus straight a with rightwards arrow on top space equals space left parenthesis straight i with hat on top space plus space 2 straight j with hat on top space plus space 6 straight k with hat on top right parenthesis space plus 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 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 space equals space straight b with rightwards arrow on top
therefore 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 are space coplanar space vectors space and space form space the space sides space of space straight a space triangle.
            straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space left parenthesis 1 right parenthesis thin space left parenthesis 2 right parenthesis space plus space left parenthesis negative 3 right parenthesis thin space left parenthesis negative 1 right parenthesis space plus space left parenthesis negative 5 right parenthesis thin space left parenthesis 1 right parenthesis space equals space 2 plus 3 space minus space 5 space equals 0
therefore space space straight a with rightwards arrow on top space and space straight b with rightwards arrow on top space are space perpendicular. therefore 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 straight a space right space angled space triangle.

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