17 M (-1; √3), N (1;-3) K (0,5; √3) Найдите: M

Найдем векторы \( \vec{KM} \) и \( \vec{KN} \):

$$\vec{KM} = (-1-0.5; \sqrt{3}-\sqrt{3}) = (-1.5; 0)$$ $$\vec{KN} = (1-0.5; -\sqrt{3}-\sqrt{3}) = (0.5; -2\sqrt{3})$$

Найдем косинус угла между векторами \( \vec{KM} \) и \( \vec{KN} \):

$$\cos{\angle M} = \frac{\vec{KM} \cdot \vec{KN}}{|\vec{KM}| \cdot |\vec{KN}|}$$ $$\vec{KM} \cdot \vec{KN} = -1.5 \cdot 0.5 + 0 \cdot (-2\sqrt{3}) = -0.75$$ $$|\vec{KM}| = \sqrt{(-1.5)^2 + 0^2} = 1.5$$ $$|\vec{KN}| = \sqrt{(0.5)^2 + (-2\sqrt{3})^2} = \sqrt{0.25 + 12} = \sqrt{12.25} = 3.5$$ $$\cos{\angle M} = \frac{-0.75}{1.5 \cdot 3.5} = \frac{-0.75}{5.25} = -\frac{1}{7}$$ $$\angle M = \arccos{-\frac{1}{7}} \approx 98.2^{\circ}$$

Ответ: \( \arccos{-\frac{1}{7}} \)