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Консультация № 189052
31.03.2016, 16:29
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Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
Найти проекцию точки А на плоскость a
А (2;-5;4) а : 5х+2y-z+3=0
Спасибо)
Найти проекцию точки А на плоскость a
А (2;-5;4) а : 5х+2y-z+3=0
Спасибо)
Обсуждение
31.03.2016, 17:01
общий
это ответ
Здравствуйте, Katerina!
Любая прямая, проходящая через заданную точку А, может быть задана уравнением
Примем в качестве координат направляющего вектора прямой координаты нормального вектора заданной плоскости [$945$]. Тогда уравнения прямой запишутся так:
Искомую проекцию найдём, решив систему уравнений
Перепишем уравнения прямой так:
и подставим в уравнение плоскости. Получим
Тогда
Ответ: A[sub][$945$][/sub]=(55/30; -152/30; 121/30).
Выкладки, разумеется, нужно проверить, чтобы избежать ошибок.
С уважением.
Любая прямая, проходящая через заданную точку А, может быть задана уравнением
(x-2)/l=(y+5)/m=(z-4)/n.
Примем в качестве координат направляющего вектора прямой координаты нормального вектора заданной плоскости [$945$]. Тогда уравнения прямой запишутся так:
(x-2)/5=(y+5)/2=(z-4)/(-1).
Искомую проекцию найдём, решив систему уравнений
5x+2y-z+3=0, (x-2)/5=(y+5)/2=(z-4)/(-1).
Перепишем уравнения прямой так:
x=5t+2, y=2t-5, z=-t+4
и подставим в уравнение плоскости. Получим
5(5t+2)+2(2t-5)-(-t+4)+3=0,
25t+10+4t-10+t-4+3=0,
30t=-1,
t=-1/30.
Тогда
x=5*(-1/30)+2=-5/30+2=55/30,
y=2*(-1/30)-5=-2/30-5=-152/30,
z=-(-1/30)+4=121/30.
Ответ: A[sub][$945$][/sub]=(55/30; -152/30; 121/30).
Выкладки, разумеется, нужно проверить, чтобы избежать ошибок.
С уважением.
5
спасибо))
Об авторе:
Facta loquuntur.
Facta loquuntur.
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