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Помогите пожалуйста с решением следующих задач: 1) На прямую 3x + 3y - 9 - 9 = 0, способную отражать лучи, падает луч 6x + 3y - 27 = 0. Составить уравнение отражённого луча. 2) Пирамида SABC задана вершинами S(3, 3, 6), A(4, -3, -3), B(-3, 4, -3), C(-3, -3, -6). Найти: а) уравнение плоскости, проходящей через точки A, B, C.; б) величину угла между ребром SA и гранью ABC; в) уравнение высоты, опущенной из вершины S на грань ABC и её длину;
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Неизвестный
18.11.2007, 00:49
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это ответ
Здравствуйте, Михайлов Дмитрий Валерьевич! 2) а) уравнение плоскости по трем точкам: | x y z 1 | | x<sub><SMALL>0</SMALL></sub> y<sub><SMALL>0</SMALL></sub> z<sub><SMALL>0</SMALL></sub> 1 | | x<sub><SMALL>1</SMALL></sub> y<sub><SMALL>1</SMALL></sub> z<sub><SMALL>1</SMALL></sub> 1 | | x<sub><SMALL>2</SMALL></sub> y<sub><SMALL>2</SMALL></sub> z<sub><SMALL>2</SMALL></sub> 1 | = | x y z 1 | | 4 -3 -3 1 | | -3 4 -3 1 | | -3 -3 -6 1 | = 3x+3y-7z-24=0
б) Уравнение прямой SA (x-x<sub><SMALL>0</SMALL></sub>)/(x<sub><SMALL>1</SMALL></sub>-x<sub><SMALL>0</SMALL></sub>) = (y-y<sub><SMALL>0</SMALL></sub>)/(y<sub><SMALL>1</SMALL></sub>-y<sub><SMALL>0</SMALL></sub>) = (z-z<sub><SMALL>0</SMALL></sub>)/(z<sub><SMALL>1</SMALL></sub>-z<sub><SMALL>0</SMALL></sub>) = (x-4)/-1 = (y+3)/6 = (z+3)/9 Угол между прямой (x-4)/-1 = (y+3)/6 = (z+3)/9 и плоскостью 3x+3y-7z-24=0 sinφ = |Al+Bm+Cn|/(√(A²+B²+C²)√(l²+m²+n²)) = |-3+18-63|/(√67√118) = 48/(√67√118)
в) Уравнения прямой, проходящей через точку (x<sub><SMALL>0</SMALL></sub>,y<sub><SMALL>0</SMALL></sub>,z<sub><SMALL>0</SMALL></sub>) перпендикулярно к плоскости Ax+By+Cz+D=0: (x-x<sub><SMALL>0</SMALL></sub>)/A = (y-y<sub><SMALL>0</SMALL></sub>)/B = (z-z<sub><SMALL>0</SMALL></sub>)/C = (x-3)/3 = (y-3)/3 = (z-6)/-7 Расстояние от точки до плоскости 3x+3y-7z-24=0 d = |Ax<sub><SMALL>0</SMALL></sub>+By<sub><SMALL>0</SMALL></sub>+Cz<sub><SMALL>0</SMALL></sub>+D|/√(A²+B²+C²) = |9+9-42-24|/√67 = 48/√67
Упростим выражения для прямых: y + x - 6 = 0 и 2x + y - 9 = 0 Во-первых найдём точку, где происходит отражение, решив систему из наших уравнений. Из первого уравнения y = 6 - x подставим во второе 2x + 6 - x - 9 = 0 => x = 3, y = 6 - x = 3. Падающий и отражённый лучи симметричны относительно нормали к прямой y + x - 6 = 0, т.е. вектору (1,1), проведённой через точку отражения. В Вашем случае эта нормаль, проведённая в точке отражения, проходит через начало координат, т.е. падающий и отражённый лучи симметричны относительно прямой y = x. Это достигается простой сменой координат в уравнении для падающего луча: 2y + x - 9 = 0 - уравнение отражённого луча.
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