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Консультация № 136816
14.05.2008, 17:14
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Уважаемые эксперты! Помогите, пожалуйста, решить задачу по высшей математике...
Задача: Выяснить компланарны ли векторы a, b и c?
а=(5,3,4); b=(4,3,3); c=(9,5,8)
Добавляйте коментарии к вычеслениям... Заранее спасибо...
Задача: Выяснить компланарны ли векторы a, b и c?
а=(5,3,4); b=(4,3,3); c=(9,5,8)
Добавляйте коментарии к вычеслениям... Заранее спасибо...
Обсуждение
Неизвестный
14.05.2008, 19:56
общий
это ответ
Здравствуйте, Umanskiy!
Чтобы векторы были компланарны, необходимо выполнение условия:
с = pa + qb
9 = 5p + 4q
5 = 3p + 3q
8 = 4p + 3q
Решаем эту систему: вычтем из 3 строки вторую
p = 3
Подставим во все строчки р = 3
4q = 9 - 15
3q = 5 - 9
3q = 8 - 12
В разных строчках получаем разные значения q, значит векторы не компланарны
Чтобы векторы были компланарны, необходимо выполнение условия:
с = pa + qb
9 = 5p + 4q
5 = 3p + 3q
8 = 4p + 3q
Решаем эту систему: вычтем из 3 строки вторую
p = 3
Подставим во все строчки р = 3
4q = 9 - 15
3q = 5 - 9
3q = 8 - 12
В разных строчках получаем разные значения q, значит векторы не компланарны
15.05.2008, 11:21
общий
это ответ
Здравствуйте, Уманский Денис!
Еще один способ проверки компланарности векторов основан на вычислении определителя.
Известно, что абсолютная величина определителя, составленного из векторов,
равна объему параллелепипеда, построенному на этих векторах. Поэтому векторы компланарны тогда и только тогда, когда определитель равен нулю.
То есть, для решения задачи достаточно найти определитель:
| 5 3 4 |
| 4 3 3 |
| 9 5 8 |.
Вычисление дает 2 - определитель не равен нулю,
значит, векторы не компланарны.
Еще один способ проверки компланарности векторов основан на вычислении определителя.
Известно, что абсолютная величина определителя, составленного из векторов,
равна объему параллелепипеда, построенному на этих векторах. Поэтому векторы компланарны тогда и только тогда, когда определитель равен нулю.
То есть, для решения задачи достаточно найти определитель:
| 5 3 4 |
| 4 3 3 |
| 9 5 8 |.
Вычисление дает 2 - определитель не равен нулю,
значит, векторы не компланарны.
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