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Vector Algebra

Question
CBSEENMA12034063
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If straight a with rightwards arrow on top space equals space 5 space straight i with hat on top space minus space straight j with hat on top space minus space space 3 space straight k with hat on top space and space straight b with rightwards arrow on top space equals space straight i with hat on top plus 3 space straight j with hat on top space minus space 5 space straight k with hat on top comma space then show that the vectors straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top space and space straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top space are space orthogonal.

Solution

       straight a with rightwards arrow on top space equals space 5 space straight i with hat on top space minus space straight j with hat on top space minus space 3 space straight k with hat on top comma space space straight b with rightwards arrow on top space equals space 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 space straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top space equals space 6 space straight i with hat on top space plus space 2 space straight j with hat on top space minus space 8 space straight k with hat on top comma space space straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top space equals space 4 space straight i with hat on top space minus space 4 space straight j with hat on top space plus space 2 space straight k with hat on top
left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis. space left parenthesis straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top right parenthesis space equals space left parenthesis 6 right parenthesis thin space left parenthesis 4 right parenthesis space plus space left parenthesis 2 right parenthesis thin space left parenthesis 4 right parenthesis space plus space left parenthesis negative 8 right parenthesis thin space left parenthesis 2 right parenthesis
                      equals space 24 minus 8 minus 16 space equals space 0
therefore space space space straight a with rightwards arrow on top plus straight b with rightwards arrow on top space and space straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top space are space orthogonal space vectors. space

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