Find the cosine of the angle between the vectors $straight a with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space 3 space straight j with hat on top space plus space 4 space straight k with hat on top space space and space straight b with rightwards arrow on top space equals 4 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 2 space straight k with hat on top$

Solution

straight a with rightwards arrow on top space equals space 2 straight i with hat on top space minus space 3 straight j with hat on top space plus space 4 straight k with hat on top comma space space straight b with rightwards arrow on top space equals space 4 straight i with hat on top plus 3 straight j with hat on top plus 2 straight k with hat on top

therefore space space straight a space equals space square root of 4 plus 9 plus 16 end root space equals space square root of 29 comma space space straight b space equals space square root of 16 plus 9 plus 4 end root space equals square root of 29

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 comma space space where space straight theta space is space angle space between space straight a with rightwards arrow on top space and space straight b with rightwards arrow on top

therefore space space left parenthesis 2 right parenthesis thin space left parenthesis 4 right parenthesis space plus space left parenthesis negative 3 right parenthesis thin space left parenthesis 3 right parenthesis space plus space left parenthesis 4 right parenthesis thin space left parenthesis 2 right parenthesis space equals space left parenthesis square root of 29 right parenthesis space left parenthesis square root of 29 right parenthesis space cos space straight theta therefore space space space space 8 space minus space 9 space plus space 8 space equals space 29 space cos space straight theta space space space space space space space space space space space rightwards double arrow space space space space space 7 space equals space 29 space cos space straight theta therefore space space space space space cos space straight theta space equals space 7 over 29

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