Aerospace Engineering & Air Traffic Control (Flight Path Conflict) Solution

Problem Statement

Two aircraft flight paths are modeled parametrically:

• Flight A: $x = 2 + 2t, y = 7t, z = 3 + 5t$
• Flight B: $x = -3 + 2s, y = -1 + 6s, z = 4 + 5s$ (distances in kilometers).

Step-by-Step Mathematical Solution

1. Direction Vector Parallelism Test

Extract direction vectors:

• Direction of Flight A: $\vec{u} = \langle 2, 7, 5 \rangle$
• Direction of Flight B: $\vec{v} = \langle 2, 6, 5 \rangle$

Compare component ratios:

• $\frac{u_x}{v_x} = \frac{2}{2} = 1$
• $\frac{u_y}{v_y} = \frac{7}{6} \approx 1.167$
• $\frac{u_z}{v_z} = \frac{5}{5} = 1$

Since $\frac{2}{2} \neq \frac{7}{6}$, the direction vectors are not parallel ($\vec{u} \neq k\vec{v}$).

2. Intersection vs. Skew Classification

Equate parametric coordinates using independent parameters $t$ and $s$:

1. $2 + 2t = -3 + 2s \implies 2t - 2s = -5 \implies t - s = -2.5$
2. $7t = -1 + 6s \implies 7t - 6s = -1$
3. $3 + 5t = 4 + 5s \implies 5t - 5s = 1 \implies t - s = 0.2$

From Equation (1), $t - s = -2.5$. From Equation (3), $t - s = 0.2$.

Because $-2.5 \neq 0.2$, this system contains an algebraic contradiction and has no simultaneous solution (inconsistent linear system).

Conclusion

Since the direction vectors are not parallel and the paths do not share any common points, the trajectories are classified as Skew lines.

Real-Life Engineering Meaning & Application

• Physical Interpretation: The flight paths never cross each other in 3D space; they exist at staggered altitudes and angles on non-coplanar planes.
• Engineering Application: NextGen Air Traffic Control (ATC) conflict-detection software executes automated skew-line verification across all en-route aircraft vectors. Confirming skew geometry verifies that the trajectories do not intersect, after which minimum clearance algorithms calculate the closest point of approach to ensure separation standards (typically 5 nautical miles lateral or 1,000 feet vertical) are maintained.