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Vector Algebra

Question
CBSEENMA12034107
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Find the angle between the vectors straight a with rightwards arrow on top space equals straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top and straight b 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 straight k with hat on top. Also find the projection of straight a with rightwards arrow on top space on space straight b with rightwards arrow on top.

Solution
straight a with rightwards arrow on top space equals 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 comma space space space straight b 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
therefore space space space space straight a space equals space square root of left parenthesis 1 right parenthesis squared plus left parenthesis 1 right parenthesis squared plus left parenthesis 1 right parenthesis squared end root space equals space square root of 3 comma space space straight b space equals space 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 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 1 right parenthesis thin space left parenthesis negative 1 right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis 1 right parenthesis space equals space 1 minus 1 plus 1 space equals space 1 Let space straight theta space be space the 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 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 space space space space space space space space rightwards double arrow space space space space 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 space space space space space rightwards double arrow space space 1 space equals space 3 space cos space straight theta therefore space space space space space space cos space straight theta space equals space 1 third space space space space space space space space space space space space space space space rightwards double arrow space space space space space straight theta space equals space cos space to the power of negative 1 end exponent space open parentheses 1 third close parentheses space space space Also space projection space of space straight a with rightwards arrow on top space on space straight b with rightwards arrow on top space space equals space fraction numerator straight a with rightwards arrow on top. space straight b with rightwards arrow on top over denominator straight b end fraction space equals fraction numerator 1 over denominator square root of 3 end fraction

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