A question about the oriented angle of two lines

Question

We know that if $d_1:a_1x+b_1y+c_1=0$ and $d_2:a_2x+b_2y+c_2=0$ then the angle between $d_1$ and $d_2$ is the angle between their normal vectors $\vec{N_1}=(a_1,b_1)$ and $\vec{N_2}=(a_2,b_2)$ and we can compute this angle by using the definition of the scalar product. But is there any way to find the oriented angle between $d_2$ and $d_1$? We are going to get that $\cos \alpha=\text{something}$ and the question is how to tell whether $\alpha$ is obtuse or not.

Answer 1

$\alpha$ is obtuse if $\alpha \gt 90 ° $. However, $cos(x)$, is negative for $x \gt 90 ° $. Therefore, $cos(\alpha) $ is negative if $\alpha$ is obtuse.