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Консультация № 198039
24.03.2020, 21:13
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Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
Написать канонические уравнения прямой X+5y-z-5= 0, 2X-5y+2z+5= 0.
Написать канонические уравнения прямой X+5y-z-5= 0, 2X-5y+2z+5= 0.
Обсуждение
29.03.2020, 10:54
общий
это ответ
Здравствуйте, master87!
Каноническое уравнение прямой в пространстве имеет вид
где (x[sub]0[/sub], y[sub]0[/sub], z[sub]0[/sub]) - произвольная точка прямой, {l, m, n} - направляющий вектор прямой. Если прямая является пересечением двух плоскостей, уравнения которых известны (как в данном случае), то в качестве направляющего вектора можно взять векторное произведение нормальных векторов этих плоскостей, которое по определению перпендикулярно обоим векторам, следовательно, параллельно плоскостям, а значит, и прямой. В качестве точки можно взять любую, чьи координаты удовлетворяют уравнениям обеих плоскостей.
Координаты нормальных векторов определим непосредственно из уравнений соответствующих плоскостей:
Их векторное произведение будет равно
то есть направляющий вектор прямой {5, -4, -15}. Координаты точки получаем, решая систему
Её решением будет, например, x = z = 0, y = 1. Тогда каноническое уравнение прямой будет иметь вид
Каноническое уравнение прямой в пространстве имеет вид
где (x[sub]0[/sub], y[sub]0[/sub], z[sub]0[/sub]) - произвольная точка прямой, {l, m, n} - направляющий вектор прямой. Если прямая является пересечением двух плоскостей, уравнения которых известны (как в данном случае), то в качестве направляющего вектора можно взять векторное произведение нормальных векторов этих плоскостей, которое по определению перпендикулярно обоим векторам, следовательно, параллельно плоскостям, а значит, и прямой. В качестве точки можно взять любую, чьи координаты удовлетворяют уравнениям обеих плоскостей.
Координаты нормальных векторов определим непосредственно из уравнений соответствующих плоскостей:
Их векторное произведение будет равно
то есть направляющий вектор прямой {5, -4, -15}. Координаты точки получаем, решая систему
Её решением будет, например, x = z = 0, y = 1. Тогда каноническое уравнение прямой будет иметь вид
5
Спасибо
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