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<p style="text-align: justify;">Qual é a distância do ponto (#a, #b, #c) para a linha recta que passa pela origem e pelo ponto (#d, #e, #f)?</p> <p style="text-align: justify;">Ao expressar o resultado use apenas duas casas decimais usando um ponto como o separador.<em><strong><br /></strong></em></p>
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<p style="text-align: justify;">A distância do ponto (#a, #b, #c) para a linha reta que passa pela origem e pelo ponto (#d, #e, #f) será o módulo de um perpendicular a linha reta e que passa pelo ponto (#a, #b, #c). Esse será o vetor «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mi»c«/mi»«mo»§#8640;«/mo»«/mover»«/math» que é dado por:</p> <p style="text-align: justify;"> «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mi»c«/mi»«mo»§#8640;«/mo»«/mover»«mo»=«/mo»«mo»(«/mo»«mo»#«/mo»«mi»a«/mi»«mo»;«/mo»«mo»§#160;«/mo»«mo»#«/mo»«mi»b«/mi»«mo»;«/mo»«mo»§#160;«/mo»«mo»#«/mo»«mi»c«/mi»«mo»)«/mo»«mo»+«/mo»«mi»k«/mi»«mo»§#183;«/mo»«mo»(«/mo»«mo»#«/mo»«mi»d«/mi»«mo»;«/mo»«mo»§#160;«/mo»«mo»#«/mo»«mi»e«/mi»«mo»;«/mo»«mo»§#160;«/mo»«mo»#«/mo»«mi»f«/mi»«mo»)«/mo»«/math».</p> <p style="text-align: justify;">A norma euclidiana (ou módulo) do vetor «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mi»c«/mi»«mo»§#8594;«/mo»«/mover»«/math» é dada por :</p> <p style="text-align: center;">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced open=¨||¨ close=¨||¨»«mover»«mi»c«/mi»«mo»§#8640;«/mo»«/mover»«/mfenced»«mo»=«/mo»«msup»«mfenced»«mrow»«munderover»«mo»§#8721;«/mo»«mrow»«mi»i«/mi»«mo»=«/mo»«mn»1«/mn»«/mrow»«mi»n«/mi»«/munderover»«msup»«mfenced open=¨|¨ close=¨|¨»«msub»«mi»c«/mi»«mi»i«/mi»«/msub»«/mfenced»«mn»2«/mn»«/msup»«/mrow»«/mfenced»«mrow»«mn»1«/mn»«mo»/«/mo»«mn»2«/mn»«/mrow»«/msup»«mo»=«/mo»«mo»#«/mo»«mi»w«/mi»«/math».</p> <p style="text-align: justify;">Fazendo-se «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»w«/mi»«mo»=«/mo»«munderover»«mo»§#8721;«/mo»«mrow»«mi»i«/mi»«mo»=«/mo»«mn»1«/mn»«/mrow»«mi»n«/mi»«/munderover»«msup»«mfenced open=¨|¨ close=¨|¨»«msub»«mi»c«/mi»«mi»i«/mi»«/msub»«/mfenced»«mn»2«/mn»«/msup»«/math», teremos:</p> <p style="text-align: center;">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»w«/mi»«mo»=«/mo»«msup»«mfenced open=¨||¨ close=¨||¨»«mover»«mi»c«/mi»«mo»§#8594;«/mo»«/mover»«/mfenced»«mn»2«/mn»«/msup»«mo».«/mo»«/math»</p> <p>O vetor «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mi»c«/mi»«mo»§#8640;«/mo»«/mover»«/math» terá um módulo mínimo quando ele for perpendicular ao vetor dado (#d, #e, #f) e que passa pela origem. Assim, o vetor «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mi»c«/mi»«mo»§#8640;«/mo»«/mover»«/math» terá módulo mínimo quando (considerando-se o valor de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»k«/mi»«/math») a derivada «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mo»d«/mo»«mi»w«/mi»«/mrow»«mrow»«mo»d«/mo»«mi»k«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mn»0«/mn»«/math», portanto:</p> <p style="text-align: center;">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mo»d«/mo»«mi»w«/mi»«/mrow»«mrow»«mo»d«/mo»«mi»k«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mo»#«/mo»«mi»w«/mi»«mo»=«/mo»«mn»0«/mn»«mo».«/mo»«/math»</p> <p style="text-align: justify;">Solucionando-se a equação em «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»k«/mi»«/math», nós acharemos que «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»k«/mi»«mo»=«/mo»«mo»#«/mo»«mi»y«/mi»«/math» (Obs.: consideraremos apenas a parte real da solução).</p> <p style="text-align: justify;">Você pode chegar a essa solução usando o comando do Matlab:</p> <p style="text-align: center;">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»s«/mi»«mi»o«/mi»«mi»l«/mi»«mi»v«/mi»«mi»e«/mi»«mo»(«/mo»«mo»`«/mo»«mo»#«/mo»«mi»w«/mi»«mo»=«/mo»«mn»0«/mn»«mo»`«/mo»«mo»)«/mo»«/math»</p> <p style="text-align: justify;"> </p> <p style="text-align: justify;">Dai:</p> <p style="text-align: center;">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mi»c«/mi»«mo»§#8640;«/mo»«/mover»«mo»=«/mo»«mo»(«/mo»«mo»#«/mo»«mi»a«/mi»«mo»;«/mo»«mo»§#160;«/mo»«mo»#«/mo»«mi»b«/mi»«mo»;«/mo»«mo»§#160;«/mo»«mo»#«/mo»«mi»c«/mi»«mo»)«/mo»«mo»+«/mo»«mi»k«/mi»«mo»§#183;«/mo»«mo»(«/mo»«mo»#«/mo»«mi»d«/mi»«mo»;«/mo»«mo»§#160;«/mo»«mo»#«/mo»«mi»e«/mi»«mo»;«/mo»«mo»§#160;«/mo»«mo»#«/mo»«mi»f«/mi»«mo»)«/mo»«mo»=«/mo»«mo»(«/mo»«mo»#«/mo»«mi»x«/mi»«mi»x«/mi»«mo»;«/mo»«mo»§#160;«/mo»«mo»#«/mo»«mi»y«/mi»«mi»y«/mi»«mo»;«/mo»«mo»§#160;«/mo»«mo»#«/mo»«mi»z«/mi»«mi»z«/mi»«mo»)«/mo»«/math»</p> <p style="text-align: center;">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced open=¨||¨ close=¨||¨»«mover»«mi»c«/mi»«mo»§#8640;«/mo»«/mover»«/mfenced»«mo»=«/mo»«mo»#«/mo»«mi»m«/mi»«/math»</p>
No, case is unimportant
Yes, case must match
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