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Здравствуйте! У меня возникли сложности с таким вопросом:
Даны координаты точек АВС. Требуется записать АВ и АС в системе орт и найти модули этих векторов. Найти угол между ними. Составить уравнение плоскости, проходящий через точку С перпендикулярно вектору АВ,где А( -2;-1;-2), В (3;0;-2),С ( 1;4;2 )
Нам требуется направляющий вектор прямой, перпендикулярную плоскость до которой мы ищем. У нас он есть? Что есть у нас? Что по условию, мы должны найти?
Об авторе:
"Если вы заметили, что вы на стороне большинства, — это верный признак того, что пора меняться." Марк Твен
Буковками i,j,k обычно обозначают единичные вектора (которые лежат на осях координат и имеют длины = 1). Буковкой k обозначают единичный вектор, лежащий на оси Oz Смотрим на нашу задачу...
Об авторе:
"Если вы заметили, что вы на стороне большинства, — это верный признак того, что пора меняться." Марк Твен
Вектор не может быть плоскостью. По условию, требуется построить уравнение плоскости, перпендикулярное вектору АВ, тем самым вектор АВ должен являться нормальным вектором плоскости (нормальным и есть перпендикулярным к плоскости)
Об авторе:
"Если вы заметили, что вы на стороне большинства, — это верный признак того, что пора меняться." Марк Твен
По Вашему диалогу с Игорем Витальевичем я понял, что учебником по аналитической геометрии Вы не пользуетесь. Отсюда все Ваши проблемы с решением задач. А ведь у Вас есть ещё линейная алгебра и математический анализ...
Я не учился ровно в два раза дольше, 32 года назад кончил ВУЗ Раз надо сейчас, значит надо брать в руки учебник и вспоминать/изучать... Включая школьный За Вас никто это не сделает...
Об авторе:
"Если вы заметили, что вы на стороне большинства, — это верный признак того, что пора меняться." Марк Твен
Чтобы у Вас была возможность при относительно небольших затратах времени повысить уровень математической культуры, предлагаю загрузить следующие книги: 1) М.И. Сканави. Элементарная математика. 2-е изд. 590 стр. djvu. 5.7 Мбздесь; 2) Д.Т. Письменный. Конспект лекций по высшей математике. Полный курс. 2009 год. 606 стр. dgvu. 3.6 Мбздесь; 3) Черненко В. Д. Высшая математика в примерах и задачах. Учебное пособие для вузов. В 3-х томах. 2003 год. Djvuздесь и систематически работать с ними.
Здравствуйте, samylovskaya1981! 1) Найдем координаты векторов АВ и АС: Как известно, AB={Xв-Xа, Yв-Yа, Zв-Za} = {3-(-2), 0-(-1), -2-(-2)} = {5, 1, 0} AC={Xc-Xa, Yc-Ya, Zc-Za} = {1-(-2), 4-(-1), 2-(-2)} = {3, 5, 4} 2) Найдем модули векторов |AB| и |AC| |AB| = [$8730$](XAB2+YAB2+ZAB2) = [$8730$](52+12+02) = [$8730$]26 |AC| = [$8730$](XAC2+YAC2+ZAC2) = [$8730$](32+52+42) = [$8730$]50 3) Найдем косинус угла между векторами cos(AB^AC) = (AB[$149$]AC)/(|AB|*|AC|) Скалярное произведение AB[$149$]AC = XAB*XAС+YAB*YAС+ZAB*ZAС = 5*3+1*5+0*4 = 20 Тогда cos(AB^AC) = 20/([$8730$]26[$8730$]50) = 20/[$8730$](26*50) = 20/[$8730$](13*2*50) = 20/[$8730$](13*102) = 20/(10*[$8730$]13) = 2/[$8730$]13 = (2[$8730$]13)/13 4) Найдем уравнение плоскости из условия равенства нулю скалярного произведения нормального (перпендикулярного) вектора к плоскости и любого вектора на плоскости, проходящего через заданную точку. По условию, вектор АВ={5;1;0} является нормальным, а плоскость проходит через точку С(1;4;2). Обозначим произвольную точку плоскости через M(x;y;z). Тогда произвольный вектор, принадлежащий плоскости и проходящий через точку С, будет иметь вид CM ={x-1;y-4;z-2} Скалярное произведение AB[$149$]CM = 5(x-1)+1(y-4)+0(z-2) = 5(x-1)+1(y-4) = 5x+y-9 Чтобы вектора АВ и СМ были перпендикулярны, необходимо, чтобы AB[$149$]CM = 0. Откуда 5x+y-9 = 0 и будет уравнением требуемой плоскости.
Об авторе:
"Если вы заметили, что вы на стороне большинства, — это верный признак того, что пора меняться." Марк Твен
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