As projeções de um ponto (C) que pertence a reta que passa pelos pontos (A) e (B), sendo: (A) [ 0; -30 ; -10] (B) [ -40; -10; 30] (C) [ .-30; ?; ?]

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\[\eqalign{ & x = {x_0} + \left( {{x_1} - {x_0}} \right)t \cr & y = {y_0} + \left( {{y_1} - {y_0}} \right)t \cr & z = {z_0} + \left( {{z_1} - {z_0}} \right)t \cr & \cr & - 30 = 0 + \left( { - 40 - 0} \right)t \cr & t = \dfrac{{ - 3}}{4} }\]

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Agora vamos encontrar as coordenadas de C, utlizando o valor de \(t\):

\[\eqalign{ & y = - 30 + \left( { - 10 + 30} \right)\left( {\dfrac{{ - 3}}{4}} \right) \cr & y = 30 - 15 \cr & y = 15 \cr & \cr & z = - 10 + \left( {30 + 10} \right)\left( {\dfrac{{ - 3}}{4}} \right) \cr & z = - 40 }\]

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Portanto, obtemos que \(\boxed{C = \left( { - 30,15, - 40} \right)}\).