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Даны точки \( \begin{array}{ll}A(-1 ; 4), B(3 ; 1), C(3 ; 4) . \\ \text { а) Найдите координаты и } \\ \text { абсолютную величину } \\ \text { вектора } \overline{A B}\end{array} \) \( \overline{C A}+\overline{A B} \) б) Найдите вектор, равный

Ask by West Barber. in Russia
Mar 12,2025

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**Ответ:** - Вектор \(\overrightarrow{AB}\) имеет координаты \((4; -3)\) и длину \(5\). - Вектор \(\overrightarrow{CA} + \overrightarrow{AB}\) равен \((0; -3)\).

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Координаты вектора \( \overline{AB} \) можно найти, вычитая координаты точки \( A \) из координат точки \( B \): \[ \overline{AB} = B - A = (3 - (-1), 1 - 4) = (4, -3). \] Теперь найдем абсолютную величину вектора \( \overline{AB} \): \[ |\overline{AB}| = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5. \] Теперь найдем координаты вектора \( \overline{CA} \): \[ \overline{CA} = A - C = (-1 - 3, 4 - 4) = (-4, 0). \] Теперь сложим векторы \( \overline{CA} + \overline{AB} \): \[ \overline{CA} + \overline{AB} = (-4, 0) + (4, -3) = (0, -3). \] Таким образом, вектор \( \overline{CA} + \overline{AB} \) равен \( (0, -3) \).

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