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Задать вопрос экспертам

Консультация № 184870

20.12.2011, 13:35

65.51 руб.

20.12.2011, 15:37

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Образование

Уважаемые эксперты! Пожалуйста, ответьте на вопрос:
Два вектора a и b имеют компоненты в условныхединицах a_x=3,2 ; a_y=1,6 ; b_x=0,5 ; b_y=4,5
Найти угол между a и b.
Найти компоненты вектора с ,который перпендикулярен а, лежит в плоскости x-y и имеет величину 5,0 y.e.

Обсуждение

давно

Гордиенко Андрей Владимирович

Мастер-Эксперт
17387
18346

20.12.2011, 14:05

общий

это ответ

Здравствуйте, Максим!

Находим скалярное произведение векторов a и b:
(a, b) = a_xb_x + a_yb_y = 3,2 [$183$] 0,5 + 1,6 [$183$] 4,5 = 8,8.

Находим абсолютные величины векторов a и b:
|a| = [$8730$](a_x² + a_y²) = [$8730$]((3,2)² + (1,6)²) = [$8730$](12,8),
|b| = [$8730$](b_x² + b_y²) = [$8730$]((0,5)² + (4,5)²) = [$8730$](20,5).

Находим угол [$966$] между векторами a и b:
[$966$] = arccos ((a, b)/(|a||b|)) = arccos (8,8/([$8730$](12,8) [$183$] [$8730$](20,5))) [$8776$] arccos 0,5433 [$8776$] 57[$186$] 6'.

Если изобразить вектор a на бумаге (начало вектора находится в начале координат, конец - в точке (3,2; 1,6)), то нетрудно убедиться, что, например вектор (1,6; -3,2) будет ему перпендикулярен. Длина этого вектора составляет
[$8730$](12,8). Тогда коллинеарный ему вектор, имеющий длину, равную 5, будет иметь пропорциональные координаты x и y, такие, что
x/1,6 = 5/[$8730$](12,8), откуда x = 5 [$183$] 1,6/[$8730$](12,8) [$8776$] 2,24,
y/(-3,2) = 5/[$8730$](12,8), откуда y = 5 [$183$] (-3,2)/[$8730$](12,8) [$8776$] -4,47.

С уважением.

Об авторе:
Facta loquuntur.

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