Intersection of line and plane (HL)

For the intersection between

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

For the point of intersection, if any, substitute the value of λ\lambda back into the line to find the coordinates. If we obtain λ\lambda can be any value, that means the line is on the plane, so the intersection is the original line.