Question

Sa se calculeze masura unghiului ascutit dintre dreptele d1:4x+3y-2=0 si d2:x+7y-4=0

Answer 1

$d_1:4x+3y-2=0 \quad \text{si}\quad d_2:x+7y-4=0 \\ \\ \text{Vectorul director dreptei }d_1 \text{ este: }\overrightarrow{v_1} = 4\vec{i}+3\vec{j} \\ \text{Vectorul director dreptei }d_2 \text{ este: }\overrightarrow{v_2} = \vec{i}+7\vec{j}$

$\begin{array}{rcl} \cos(\overrightarrow{v_1}, \overrightarrow{v_2}) &=& \dfrac{\overrightarrow{v_1} \cdot \overrightarrow{v_2}}{|\overrightarrow{v_1}|\cdot |\overrightarrow{v_2}|} \\ \\&=& \dfrac{4\cdot 1 + 3\cdot 7}{\sqrt{4^2+3^2}\cdot \sqrt{1^2+7^2}} \\ \\ &=& \dfrac{25}{\sqrt{25}\cdot\sqrt{50} } \\ \\ &=& \dfrac{25}{5\cdot 5\sqrt 2} \\ \\ &=& \dfrac{1}{\sqrt 2} \\ \\ &=& \dfrac{\sqrt{2}}{2}\end{array}$

$\Rightarrow \cos(\overrightarrow{v_1}, \overrightarrow{v_2}) = \dfrac{\sqrt 2}{2} \\ \\ \text{Unghiul dintre cele 2 drepte este acelasi cu unghiul dintre} \\ \text{cei 2 vectori.} \\ \\ \Rightarrow \text{Unghiul dintre cele 2 drepte este: } \\ \arccos\Big(\dfrac{\sqrt 2}{2}\Big) = \dfrac{\pi}{4} = 45^{\circ}$