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Консультация № 173896
01.11.2009, 18:50
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Вечер добрый, многоуважаемые эксперты!!!! 
Помогите с рещением небольшой задачки
Прямая проходит через две точки М(7, -2, 1) и N (3, 0, -6). Найти угол её наклона к плоскости, отсекающей на осях отрезки, равные 4.
Заранее говорю спасибо, в надежде что вы мне поможите!
Помогите с рещением небольшой задачки
Прямая проходит через две точки М(7, -2, 1) и N (3, 0, -6). Найти угол её наклона к плоскости, отсекающей на осях отрезки, равные 4.
Заранее говорю спасибо, в надежде что вы мне поможите!
Обсуждение
05.11.2009, 21:06
общий
это ответ
Здравствуйте, Alexxxx 23.
Составим уравнение заданной прямой:
(x – 7)/(3 – 7) = (y – (-2))/(0 – (-2)) = (z – 1)/(-6 – 1),
(x – 4)/(-4) = (y + 2)/2 = (z – 1)/(-7).
Поскольку заданная плоскость, согласно условию, отсекает на координатных осях отрезки, равные 4, постольку ее уравнение в отрезках имеет следующий вид:
x/4 + y/4 + z/4 = 1,
а общее уравнение суть
x + y + z – 1 = 0.
Воспользовавшись известной формулой для нахождения угла между прямой и плоскостью, находим искомый угол:
sin φ = |1 ∙ (-4) + 1 ∙ 2 + 1 ∙ (-7)|/(√((-4)2 + 22 + (-7)2) ∙ √(12 + 12 + 12)) = 9/(√69 ∙ √3) ≈ 0,6255,
φ ≈ 38° 43’.
Ответ: 38° 43’.
С уважением.
Составим уравнение заданной прямой:
(x – 7)/(3 – 7) = (y – (-2))/(0 – (-2)) = (z – 1)/(-6 – 1),
(x – 4)/(-4) = (y + 2)/2 = (z – 1)/(-7).
Поскольку заданная плоскость, согласно условию, отсекает на координатных осях отрезки, равные 4, постольку ее уравнение в отрезках имеет следующий вид:
x/4 + y/4 + z/4 = 1,
а общее уравнение суть
x + y + z – 1 = 0.
Воспользовавшись известной формулой для нахождения угла между прямой и плоскостью, находим искомый угол:
sin φ = |1 ∙ (-4) + 1 ∙ 2 + 1 ∙ (-7)|/(√((-4)2 + 22 + (-7)2) ∙ √(12 + 12 + 12)) = 9/(√69 ∙ √3) ≈ 0,6255,
φ ≈ 38° 43’.
Ответ: 38° 43’.
С уважением.
Об авторе:
Facta loquuntur.
Facta loquuntur.
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