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Здравствуйте! У меня возникли сложности с таким вопросом: Первая прямая проходит через точки A=(-3;7;-3) и B=(-3;8;-3). Вторая прямая проходит через точки C=(-7;2;-5) и D=(-9;2;-6). Найти координаты точки пересечения этих прямых. Ответ запишите в виде "(12;-34;56)". Без пробелов. Посмотрите пожалуйста, не могу никак решить
Просто обратите внимание, что у первой прямой x=-3 и z=-3, а у второй y=2 - поэтому точкой пересечения может быть только (-3;2;-3)
Впрочем, нужно ещё убедиться, что вторая прямая проходит через эту точку. Для этого найдём для неё зависимость остальных двух координат. x=2z+3 Да, найденная точка действительно ей удовлетворяет
Согласен, этот вариант решения подошел. Но если числа буду разные. Например: Первая прямая проходит через точки A=(7;-6;3) и B=(8;-7;3). Вторая прямая проходит через точки C=(-4;3;9) и D=(-8;6;12). Найти координаты точки пересечения этих прямых. Ответ запишите в виде "(12;-34;56)". Без пробелов.
В общем виде, прямая в трёхмерном пространстве задаётся двумя уравнениями вида y=ax+b и z=cx+d
Сперва берём одну пару координат, например x и y, и решаем систему уравнений обеих прямых в этих координатах. Затем подставляем найденные координаты в уравнения третьей координаты - если для обеих прямых она совпадает, то мы нашли точку пересечения, в противном случае это скрещивающиеся прямые
В приведённом Вами примере уравнения прямых y=-x+1; z=3 y=-3x/4; z=-3x/4+6
Приравниваем y(x) -x+1=-3x/4 4x-4=3x x=4 y=-3
Находим z для первой прямой z=3 для второй прямой z=-3×4/4+6=3 координаты совпали, поэтому точка пересечения (4;-3;3)
Прямая в пространстве задаётся двумя уравнениями, которые соответствуют её проекциям на координатные плоскости. А находятся они из координат точек действительно так же, как и уравнения прямой на плоскости - берёте координаты попарно и находите их зависимость
Первая прямая проходит через точки A=(2;3;5) и B=(2;2;6). Вторая прямая проходит через точки C=(5;7;1) и D=(6;8;0). Найти координаты точки пересечения этих прямых. Ответ запишите в виде "(12;-34;56)". Без пробелов. Решил эту задачу вашим способом. Получилось х=1, y=3, z=5 в обоих уравнениях z = 5. Но программа показывает, что ответ не правильный. Записал во так: (1;3;5). Нашёл а и b в обоих уравнениях. Приравнял правые части, нашёл х и y. Потом z в обоих уравнениях. Где я ошибся?
Здравствуйте, maksi941! Просто обратите внимание, что у первой прямой x=-3 и z=-3, а у второй y=2 - поэтому точкой пересечения может быть только (-3;2;-3)
Впрочем, нужно ещё убедиться, что вторая прямая проходит через эту точку. Для этого найдём для неё зависимость остальных двух координат. x=2z+3 Да, найденная точка действительно ей удовлетворяет
В общем виде, прямая в трёхмерном пространстве задаётся двумя уравнениями вида y=ax+b и z=cx+d
Сперва берём одну пару координат, например x и y, и решаем систему уравнений обеих прямых в этих координатах. Затем подставляем найденные координаты в уравнения третьей координаты - если для обеих прямых она совпадает, то мы нашли точку пересечения, в противном случае это скрещивающиеся прямые
В приведённом Вами в мини-форуме примере
уравнения прямых: y=-x+1; z=3 y=-3x/4; z=-3x/4+6
Приравниваем y(x) -x+1=-3x/4 4x-4=3x x=4 y=-3
Находим z для первой прямой z=3 для второй прямой z=-3×4/4+6=3 координаты совпали, поэтому точка пересечения (4;-3;3)
Уравнения первой прямой x=2; z=8-y второй прямой y=x+2; z=6-x Так что x=1 тут никак не выходит. Зато подставляя x=2 в уравнения второй прямой, сразу получаем ответ (2;4;4)
Тут главное не запутаться в порядке координат, особенно в таких частных случаях
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