Introduction

In this fourth unit on vectors, we have learned three common ways to express the equation of a plane in \(\mathbb{R}^3\).

These are vector, parametric, and scalar equations of the plane.

In this module, we will consider how to determine if a given line intersects a given plane.

Given a line and a plane in \(\mathbb{R}^3\), there are three possibilities for the intersection of the line with the plane.

1. The line and the plane intersect at a single point. There is exactly one solution.
2. The line is parallel to the plane. The line and the plane do not intersect. There are no solutions.
3. The line lies on the plane, so every point on the line intersects the plane. There are an infinite number of solutions.

To determine algebraically whether or not a line intersects a plane, we may substitute the parametric equations of the line into the scalar equation of the plane.

The resulting linear equation reduces to one of three possibilities, with each determining the type of intersection between the line and the plane.