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Vector Algebra

Question
CBSEENMA12034116
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Prove that the vectors
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 straight k with hat on top comma space space space space space straight b with rightwards arrow on top space equals space straight i with hat on top space minus 3 space straight j with hat on top space minus space 5 space straight k with hat on top comma space space straight c with rightwards arrow on top space equals space 3 space straight i with hat on top space minus space 4 space straight j with hat on top space minus space 4 space straight k with hat on top forms the sides of a right-angled triangle. Also, find the remaining angles of the triangle.

Solution
Here
             straight a with rightwards arrow on top equals space 2 straight i with hat on top minus straight j with hat on top plus straight k with hat on top comma space straight b with rightwards arrow on top equals straight i with hat on top space minus 3 space straight j with hat on top space minus space 5 space straight k with hat on top comma space space straight c with rightwards arrow on top space equals space 3 space straight i with hat on top space minus space 4 space straight j with hat on top space minus space 4 space straight k with hat on top
Now,    straight a with rightwards arrow on top plus straight b with rightwards arrow on top space equals space left parenthesis 2 straight i with hat on top minus straight j with hat on top plus straight k with hat on top right parenthesis plus left parenthesis straight i with hat on top minus 3 space straight j with hat on top minus 5 space straight k with hat on top right parenthesis equals 3 space straight i with hat on top minus 4 space straight j with hat on top minus 4 space straight k with hat on top
therefore space space straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top space are space coplanar space vectors space and space form space the space sides space of space straight a space triangle.
Now,   straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space left parenthesis 2 right parenthesis space left parenthesis 1 right parenthesis space plus space left parenthesis negative 1 right parenthesis thin space left parenthesis negative 3 right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis negative 5 right parenthesis space equals space 2 plus 3 minus 5 space equals space 0

therefore space space space space straight a with rightwards arrow on top space and space straight b with rightwards arrow on top space are space perpendicular therefore space space space straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top space form space the space sides space of space straight a space right minus angled space triangle. space space space space space space space space space space space space cos space straight A space equals space fraction numerator straight b with rightwards arrow on top. straight c with rightwards arrow on top over denominator open vertical bar straight b with rightwards arrow on top close vertical bar. space open vertical bar straight c 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 1 right parenthesis left parenthesis 3 right parenthesis plus left parenthesis negative 3 right parenthesis thin space left parenthesis negative 4 right parenthesis space plus space left parenthesis negative 5 right parenthesis thin space left parenthesis negative 4 right parenthesis over denominator square root of 1 plus 9 plus 25 end root space square root of 9 plus 16 plus 6 end root 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 3 plus 12 plus 20 over denominator square root of 35 space square root of 41 end fraction space equals fraction numerator 35 over denominator square root of 35 space square root of 41 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 fraction numerator square root of 35 over denominator square root of 41 end fraction space equals square root of 35 over 41 end root therefore space space space space space space space straight A space equals space cos to the power of negative 1 end exponent open parentheses square root of 35 over 41 end root close parentheses space space space space space space space space space cos space straight B space equals space fraction numerator straight c with rightwards arrow on top. space straight a with rightwards arrow on top over denominator open vertical bar straight c with rightwards arrow on top close vertical bar space open vertical bar straight a with rightwards arrow on top close vertical bar end fraction space equals fraction numerator left parenthesis 3 right parenthesis left parenthesis 2 right parenthesis plus left parenthesis negative 4 right parenthesis thin space left parenthesis negative 1 right parenthesis plus left parenthesis negative 4 right parenthesis thin space left parenthesis 1 right parenthesis over denominator square root of 9 plus 16 plus 16 end root space square root of 4 plus 1 plus 1 end root end fraction space space space space space space space space space space space space space space space space space space space space equals space fraction numerator 6 plus 4 minus 4 over denominator square root of 41 space square root of 6 end fraction space equals space fraction numerator square root of 6 over denominator square root of 41 space square root of 6 end fraction space equals space fraction numerator square root of 6 over denominator square root of 41 end fraction equals space square root of 6 over 41 end root because space space space space straight B space equals space cos to the power of negative 1 end exponent space open parentheses square root of 6 over 41 end root close parentheses




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