Skew drone paths and the unchecked coordinate
Two drone flight paths are modelled by straight lines in a three-dimensional coordinate system measured in kilometres. Path L1 has equation r=(1,2,3)+lambda(2,-1,1); path L2 has equation r=(4,0,5)+mu(1,1,2). A flight-control student sets the x- and y-coordinates equal, solves 1+2lambda=4+mu and 2-lambda=mu, obtaining lambda=5/3 and mu=1/3. The student concludes that the paths intersect at the point (1+2(5/3), 2-5/3, 3+5/3)=(13/3,1/3,14/3). A supervisor notes that the z-coordinates have not been checked and that the two aircraft could pass at different heights. The team must determine the true geometric relationship between the paths and, if they are skew, find the shortest distance between them so that a safety exclusion zone can be set.
1. Show that the two paths do not intersect. State whether they are parallel, intersecting or skew, and explain the significance of the unused z-equation.
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Equating x and y gave lambda=5/3 and mu=1/3. For these parameters the z-coordinate on L1 is 3+5/3=14/3, while on L2 it is 5+2(1/3)=17/3. These are not equal, so no single pair (lambda,mu) satisfies all three coordinate equations. The direction vectors (2,-1,1) and (1,1,2) are not scalar multiples, so the lines are not parallel. Since they are not parallel and do not intersect, they are skew. The x,y solution is only a candidate from a two-equation projection; the unused z-equation is the check that decides intersection. The student's point lies on L1 but not on L2.
2. Find a vector perpendicular to both direction vectors and use it to calculate the shortest distance between the two paths. Interpret the result in the drone context.
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A normal vector is n=d1 x d2=(2,-1,1)x(1,1,2)=(-3,-3,3), or simplify to (-1,-1,1). Take A=(1,2,3) on L1 and B=(4,0,5) on L2, so AB=(3,-2,2). The shortest distance between skew lines is |AB dot n|/|n|. Using n=(-1,-1,1): AB dot n=3(-1)+(-2)(-1)+2(1)=-3+2+2=1, and |n|=sqrt(3). Hence distance=1/sqrt(3)≈0.577 km, about 577 m. Using the unsimplified n gives 3/(3sqrt3), the same value. The minimum geometric separation of these infinite path lines is about 0.58 km. A real operational separation requirement must be supplied separately, and line geometry alone does not establish simultaneous aircraft positions, a safe exclusion radius, or an aviation operating decision.