AI Tutor
When two aircraft cross paths on a radar screen, locating the intersection coordinate is simple algebraβ€”but determining their exact collision or bypass angle requires measuring the angle between two straight lines.
Every non-vertical line makes an angle of inclination α\alpha with the positive xx-axis, where the slope equals the tangent of that angle: m=tan⁑(α)m = \tan(\alpha).
How can we connect the inclination angles of two different lines to find the angle where they cross?

The Geometry Behind the Intersection

When two lines L1L_1 and L2L_2 intersect, they form a triangle with the xx-axis. The inclination angles are Ξ±1\alpha_1 and Ξ±2\alpha_2, and the angle between the lines is ΞΈ\theta.
By the exterior angle theorem, the outer angle Ξ±2\alpha_2 equals the sum of the two opposite interior angles: Ξ±2=Ξ±1+ΞΈ\alpha_2 = \alpha_1 + \theta, which rearranges to ΞΈ=Ξ±2βˆ’Ξ±1\theta = \alpha_2 - \alpha_1.
In the 10th century, Persian mathematician AbΕ« al-Wafā al-BΕ«zjānΔ« formalized angle difference identities, establishing that tan⁑(Ξ±2βˆ’Ξ±1)=tan⁑α2βˆ’tan⁑α11+tan⁑α1tan⁑α2\tan(\alpha_2 - \alpha_1) = \frac{\tan\alpha_2 - \tan\alpha_1}{1 + \tan\alpha_1\tan\alpha_2}.
Substituting slopes m1=tan⁑α1m_1 = \tan\alpha_1 and m2=tan⁑α2m_2 = \tan\alpha_2, and applying absolute value bars to guarantee the acute angle (between 0∘0^\circ and 90∘90^\circ), gives our formula.
How do we apply this formula to solve an actual pair of equations step-by-step?

Step-by-Step Calculation