Question Video: Understanding the Relation between a Plane and a Parallel Line Mathematics

Video Transcript

Complete: If line 𝐿 is parallel to the plane 𝑋, then line 𝐿 intersection 𝑋 equals what.

It might be helpful to begin this question with a diagram of the line 𝐿 parallel to a plane 𝑋. So here is a plane 𝑋, which extends infinitely in all directions, and here is the line 𝐿. Now, before we consider how to fill in the gap in this statement, let’s consider the different relationships we can have between lines and planes in space. Let’s consider a different plane. We can call this plane plane 𝐴. Then, the first kind of relationship we can have between a line and a plane is illustrated by this line 𝐵. Line 𝐵 lies on the plane. This means that every point on the line 𝐵 also lies on the plane 𝐴.

The second type of relationship we can have is illustrated by this line 𝐶. Line 𝐶 intersects the plane. So this means that the line and the plane share a common point. Another special type of a line which intersects a plane is one which does so at right angles, in which case we would say that the line intersects the plane orthogonally. The final type of relationship we have is when we have a line which is parallel to the plane. This was illustrated in the first diagram.

Now that we have seen the different relationships between lines and planes, this means we have demonstrated that if a line does not intersect a plane, the line is parallel to the plane. So when we are considering the intersection of the line 𝐿 and the plane 𝑋, there will be no intersection. And therefore to complete this statement using set notation, we can use the symbol for the empty or null set.