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Vector Algebra

Question
CBSEENMA12034061
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Find the angle between vectors straight a with rightwards arrow on top plus straight b with rightwards arrow on top and straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top if straight a with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space straight j with hat on top space plus space 3 space straight k with hat on top and straight b with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space straight j with hat on top space minus space 2 space straight k with hat on top

Solution
straight a with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space straight j with hat on top space plus space 3 space straight k with hat on top comma space space straight b with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space straight j with hat on top space minus space 2 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 5 space straight i with hat on top space plus 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 minus straight i with hat on top space minus space space 2 space straight j with hat on top space plus space 5 space straight k with hat on top space space space space space space space space space space open vertical bar straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top close vertical bar space equals space square root of 25 plus 1 end root space equals space square root of 26 space space space space space space space space space space space open vertical bar straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top close vertical bar space equals space square root of 1 plus 4 plus 25 end root space equals space square root of 30 Let space straight theta space space be space angle space between space straight a with rightwards arrow on top plus space straight b with rightwards arrow on top space and space straight a with rightwards arrow on top space space minus space straight b with rightwards arrow on top. therefore space space space space space cos space straight theta space space space equals space space fraction numerator left parenthesis straight a with rightwards arrow on top space plus space 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 over denominator open vertical bar straight a with rightwards arrow on top plus space straight b with rightwards arrow on top close vertical bar. space open vertical bar straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top close vertical bar end fraction space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator left parenthesis 5 right parenthesis thin space left parenthesis negative 1 right parenthesis space plus space left parenthesis 0 right parenthesis thin space left parenthesis negative 2 right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis 5 right parenthesis over denominator square root of 26. space square root of 30 end fraction space equals space fraction numerator negative 5 plus 0 plus 5 over denominator square root of 26 space square root of 30 end fraction therefore space space space space space cos space straight theta space equals space 0 space space space space rightwards double arrow space space space straight theta space equals space 90 degree. space

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