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Vector Algebra

Question
CBSEENMA12034109
Wired Faculty App

Find the cosine of the angle between the vectors
straight a with rightwards arrow on top space equals space 5 space straight i with hat on top space plus 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 minus 2 space straight i with hat on top space plus space straight j with hat on top space plus space 3 space straight k with hat on top

Solution

Here,   straight a with rightwards arrow on top space equals space 5 straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top comma space space space straight b with rightwards arrow on top space equals negative 2 space straight i with hat on top space plus space straight j with hat on top space plus space 3 space straight k with hat on top
therefore space space space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space left parenthesis 5 straight i with hat on top plus straight j with hat on top plus straight k with hat on top right parenthesis. space left parenthesis negative 2 straight i with hat on top space plus straight j with hat on top plus 3 space straight k with hat on top right parenthesis space space space space space space space space space space space space space space space space space equals space left parenthesis 5 right parenthesis thin space left parenthesis negative 2 right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis 1 right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis 3 right parenthesis space equals negative 10 plus 1 plus 3 space equals space minus 6
Let straight theta be angle between straight a with rightwards arrow on top space and space straight b with rightwards arrow on top.
therefore space space space 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
   rightwards double arrow space space space minus 6 space equals space square root of 5 plus 1 plus 1 end root space square root of 4 plus 1 plus 9 end root space cos space straight theta rightwards double arrow space space minus 6 space equals space square root of 7 space square root of 14 space cos space straight theta space space rightwards double arrow space space space space minus 6 space equals space 7 square root of 2 space cos space straight theta therefore space space space cos space straight theta space equals space minus fraction numerator 6 over denominator 7 square root of 2 end fraction

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