990 Даны векторы \(\overrightarrow{a} \{3; 4\}, \overrightarrow{b} \{6; -8\}, \overrightarrow{c} \{1; 5\}\). a) Найдите координаты векторов \(\overrightarrow{p}=\overrightarrow{a}+\overrightarrow{b}\), \(\overrightarrow{q}=\overrightarrow{b}+\overrightarrow{c}\), \(\overrightarrow{r}=2\overrightarrow{a}-\overrightarrow{b}+\overrightarrow{c}\), \(\overrightarrow{s}=\overrightarrow{a}-\overrightarrow{b}-\overrightarrow{c}\). б) Найдите \(|\overrightarrow{a}|, |\overrightarrow{b}|, |\overrightarrow{p}|, |\overrightarrow{q}|\).

Ответ: a) \(\overrightarrow{p} \{9; -4\}, \overrightarrow{q} \{7; -3\}, \overrightarrow{r} \{1; 21\}, \overrightarrow{s} \{-4; 7\}\); б) \(|\overrightarrow{a}|=5, |\overrightarrow{b}|=10, |\overrightarrow{p}|=\sqrt{97}, |\overrightarrow{q}|=\sqrt{58}\)

Решение:

a) Найдите координаты векторов \(\overrightarrow{p}=\overrightarrow{a}+\overrightarrow{b}\), \(\overrightarrow{q}=\overrightarrow{b}+\overrightarrow{c}\), \(\overrightarrow{r}=2\overrightarrow{a}-\overrightarrow{b}+\overrightarrow{c}\), \(\overrightarrow{s}=\overrightarrow{a}-\overrightarrow{b}-\overrightarrow{c}\).

Находим координаты векторов \(\overrightarrow{p}, \overrightarrow{q}, \overrightarrow{r}, \overrightarrow{s}\):

\[\overrightarrow{p} = \overrightarrow{a} + \overrightarrow{b} = (3+6; 4-8) = (9; -4)\]

\[\overrightarrow{q} = \overrightarrow{b} + \overrightarrow{c} = (6+1; -8+5) = (7; -3)\]

\[\overrightarrow{r} = 2\overrightarrow{a} - \overrightarrow{b} + \overrightarrow{c} = (2\cdot3-6+1; 2\cdot4-(-8)+5) = (6-6+1; 8+8+5) = (1; 21)\]

\[\overrightarrow{s} = \overrightarrow{a} - \overrightarrow{b} - \overrightarrow{c} = (3-6-1; 4-(-8)-5) = (-4; 7)\]

б) Найдите \(|\overrightarrow{a}|, |\overrightarrow{b}|, |\overrightarrow{p}|, |\overrightarrow{q}|\).

Находим длины векторов \(\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{p}, \overrightarrow{q}\):

\[|\overrightarrow{a}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\]

\[|\overrightarrow{b}| = \sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10\]

\[|\overrightarrow{p}| = \sqrt{9^2 + (-4)^2} = \sqrt{81 + 16} = \sqrt{97}\]

\[|\overrightarrow{q}| = \sqrt{7^2 + (-3)^2} = \sqrt{49 + 9} = \sqrt{58}\]

Ответ: a) \(\overrightarrow{p} \{9; -4\}, \overrightarrow{q} \{7; -3\}, \(\overrightarrow{r} \{1; 21\}, \overrightarrow{s} \{-4; 7\}\); б) \(|\overrightarrow{a}|=5, |\overrightarrow{b}|=10, |\overrightarrow{p}|=\sqrt{97}, |\overrightarrow{q}|=\sqrt{58}\)