1077 Найдите координаты вектора \(\overrightarrow{p}\) и его длину, если: a) \(\overrightarrow{p}=7\overrightarrow{a}-3\overrightarrow{b}\), \(\overrightarrow{a}\) {1; -1}, \(\overrightarrow{b}\) {5; -2}; б) \(\overrightarrow{p}=4\overrightarrow{a}-2\overrightarrow{b}\), \(\overrightarrow{a}\) {6; 3}, \(\overrightarrow{b}\) {5; 4}; в) \(\overrightarrow{p}=5\overrightarrow{a}-4\overrightarrow{b}\), \(\overrightarrow{a}\) {\(\frac{3}{5}\); \(\frac{1}{5}\)}, \(\overrightarrow{b}\) {6; -1}; г) \(\overrightarrow{p}=3(-2\overrightarrow{a}-4\overrightarrow{b})\), \(\overrightarrow{a}\) {1; 5}, \(\overrightarrow{b}\) {-1; -1}.

a) \(\overrightarrow{p}=7\overrightarrow{a}-3\overrightarrow{b}\), \(\overrightarrow{a}\) {1; -1}, \(\overrightarrow{b}\) {5; -2} $$\overrightarrow{p} = 7(1; -1) - 3(5; -2) = (7; -7) - (15; -6) = (7-15; -7-(-6)) = (-8; -1)$$ Длина вектора: $$|\overrightarrow{p}| = \sqrt{(-8)^2 + (-1)^2} = \sqrt{64+1} = \sqrt{65}$$

б) \(\overrightarrow{p}=4\overrightarrow{a}-2\overrightarrow{b}\), \(\overrightarrow{a}\) {6; 3}, \(\overrightarrow{b}\) {5; 4} $$\overrightarrow{p} = 4(6; 3) - 2(5; 4) = (24; 12) - (10; 8) = (24-10; 12-8) = (14; 4)$$ Длина вектора: $$|\overrightarrow{p}| = \sqrt{(14)^2 + (4)^2} = \sqrt{196+16} = \sqrt{212} = 2\sqrt{53}$$

в) \(\overrightarrow{p}=5\overrightarrow{a}-4\overrightarrow{b}\), \(\overrightarrow{a}\) {\(\frac{3}{5}\); \(\frac{1}{5}\)}, \(\overrightarrow{b}\) {6; -1} $$\overrightarrow{p} = 5(\frac{3}{5}; \frac{1}{5}) - 4(6; -1) = (3; 1) - (24; -4) = (3-24; 1-(-4)) = (-21; 5)$$ Длина вектора: $$|\overrightarrow{p}| = \sqrt{(-21)^2 + (5)^2} = \sqrt{441+25} = \sqrt{466}$$

г) \(\overrightarrow{p}=3(-2\overrightarrow{a}-4\overrightarrow{b})\), \(\overrightarrow{a}\) {1; 5}, \(\overrightarrow{b}\) {-1; -1} $$\overrightarrow{p} = 3(-2(1; 5) - 4(-1; -1)) = 3((-2; -10) - (-4; -4)) = 3((-2+4; -10+4)) = 3(2; -6) = (6; -18)$$ Длина вектора: $$|\overrightarrow{p}| = \sqrt{(6)^2 + (-18)^2} = \sqrt{36+324} = \sqrt{360} = 6\sqrt{10}$$

Ответ: a) \(\overrightarrow{p}\) {-8; -1}, |\(\overrightarrow{p}\)| = \(\sqrt{65}\) b) \(\overrightarrow{p}\) {14; 4}, |\(\overrightarrow{p}\)| = \(2\sqrt{53}\) в) \(\overrightarrow{p}\) {-21; 5}, |\(\overrightarrow{p}\)| = \(\sqrt{466}\) г) \(\overrightarrow{p}\) {6; -18}, |\(\overrightarrow{p}\)| = \(6\sqrt{10}\)