Recall that, given a line and a plane in \(\mathbb{R}^3\), there are three possibilities for the intersection of the line with the plane.
- The line is parallel to the plane (orthogonal to a normal vector of the plane). The line and the plane do not intersect. There are no solutions.
- The line and the plane intersect at a single point. There is exactly one solution.
- The line lies in the plane, so every point on the line intersects the plane. There are an infinite number of solutions.