Question

Find the angle between the vectors $straight a with rightwards arrow on top space equals straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top space$ and $straight b with rightwards arrow on top space equals space straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top.$

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Solution

Let

straight theta

be angle between the vectors

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 space plus 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 space plus space straight j with hat on top space minus space straight k with hat on top

Now,

straight a space equals square root of left parenthesis 1 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 1 plus 1 plus 1 end root space equals square root of 3

straight b equals space square root of left parenthesis 1 right parenthesis squared plus left parenthesis 1 right parenthesis squared plus left parenthesis negative 1 right parenthesis squared end root space equals square root of 1 plus 1 plus 1 end root space equals square root of 3

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 1 right parenthesis space plus space left parenthesis negative 1 right parenthesis thin space left parenthesis 1 right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis negative 1 right parenthesis space equals space 1 minus 1 minus 1 space equals negative 1

Now

straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space straight a space straight b space cos space straight theta space space space rightwards double arrow space space space space minus 1 space equals space left parenthesis square root of 3 right parenthesis thin space left parenthesis square root of 3 right parenthesis space cos space straight theta

therefore space space space space space 3 space cos space straight theta space equals space minus 1 space space space space space space space space space or space space space cos space straight theta space equals space minus 1 third therefore space space space space space space space space space straight theta space equals space cos to the power of negative 1 end exponent space open parentheses negative 1 third close parentheses.

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