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Консультация № 198037
24.03.2020, 21:12
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Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
Вершины пирамиды находятся в точках А(-4,-5,-3), В(3,1,2), С(5,7,-6), D(6,-1,5). Найти объем пирамиды и длину высоты, опущенной из вершины В.
Вершины пирамиды находятся в точках А(-4,-5,-3), В(3,1,2), С(5,7,-6), D(6,-1,5). Найти объем пирамиды и длину высоты, опущенной из вершины В.
Обсуждение
29.03.2020, 07:17
общий
это ответ
Здравствуйте, master87!
Пусть имеются три вектора a, b, c с общим началом и не лежащие в одной плоскости (некомпланарные). Тогда объём параллелепипеда, построенного на этих векторах, будет равен модулю их смешанного произведения, объём же пирамиды будет в шесть раз меньше:
Для параллелепипеда площадь основания - параллелограмма, построенного на векторах b и c, равна модулю их векторного произведения, для пирамиды площадь лежащего в основании треугольника будет вдвое меньше:
Поскольку для пирамиды V = SH/3, её высота будет равна
В данном случае в качестве векторов a, b, c можно взять соответственно вектора
Тогда соответствующие смешанное и векторное произведения определяются по формулам векторной алгебры как
а искомые объём и высота пирамиды будут равны
Пусть имеются три вектора a, b, c с общим началом и не лежащие в одной плоскости (некомпланарные). Тогда объём параллелепипеда, построенного на этих векторах, будет равен модулю их смешанного произведения, объём же пирамиды будет в шесть раз меньше:
Для параллелепипеда площадь основания - параллелограмма, построенного на векторах b и c, равна модулю их векторного произведения, для пирамиды площадь лежащего в основании треугольника будет вдвое меньше:
Поскольку для пирамиды V = SH/3, её высота будет равна
В данном случае в качестве векторов a, b, c можно взять соответственно вектора
Тогда соответствующие смешанное и векторное произведения определяются по формулам векторной алгебры как
а искомые объём и высота пирамиды будут равны
5
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