Intersection of line and plane (HL)

For the intersection between the line

r=(x0,y0,z0)+λ(l,m,n)\bm r = (x_0, y_0, z_0) + \lambda(l, m, n)

and the plane

ax+by+cz=dax+by+cz=d

There are 3 cases:

  1. Line is on the plane (infinite intersections).
  2. Line is on a parallel plane (no intersections)
  3. Intersection (exactly one intersection)

We can substitute the parametric equations of a line into equation of the plane.

a(lλ+x0)+b(mλ+y0)+c(nλ+z0)=da(l\lambda+x_0) + b(m\lambda+y_0) + c(n\lambda+z_0) = d

It’s a one-variable linear equation. The number of solutions of λ\lambda corresponds to the number of intersections between the line and the plane. Namely, no solution for λ\lambda means no solution; one solution for λ\lambda means a single intersection point; λ\lambda can be any value means the line is on the plane.

For the intersection point, if any, substitute the value of λ\lambda back into the line to find the coordinates

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