Lesson Video: Points, Lines, and Planes in Space

Video Transcript

In this video, we will learn to identify and model geometric concepts, points, lines, and planes in space. We’ll also consider their properties and what happens when they interact with one another. Before we start, let’s remind ourselves what we mean when we say points, lines, and planes.

In geometry, a point is a location. It has neither shape nor size. However, when we’re working with the concept of a point, we have to represent it somehow. And so, we represent it with a dot, and it’s generally named by a capital letter. If you saw something like this, you would call it the point 𝐴. In algebra, we consider the location of points in two dimensions, along the 𝑥- and 𝑦-axis, which would give point 𝐴 an 𝑥-coordinate and a 𝑦-coordinate. But when we’re talking about a point in space, it has a third dimension of 𝑧, which means that the coordinates of 𝐴 in space will be made up of three components: an 𝑥, a 𝑦, and a 𝑧. For the purposes of this video, we won’t be labeling our points, but it’s good to remember that those do exist.

A line is a straight set of points that extend infinitely in two directions. If we have two points 𝐴 and 𝐵, there is exactly one line that passes through both points. Any other routes to get from 𝐴 to 𝐵 would have to include a curve. We use arrows on either end of the line to indicate that it extends infinitely in two directions. One way to name a line is by giving them a name of any two points that fall on that line. In this case, we could call this line 𝐴𝐵.

Sometimes, lines are named with a variable. When this happens, they’re often lowercase script letters. And then, this line could be called line 𝑚. The written notation for the line 𝐴𝐵 would be the capital letters 𝐴𝐵 with a small line above it. It would also be correct to write line 𝑚. If we’re specifically talking about the space from point 𝐴 to point 𝐵, it does not fit the definition of a line, and that is why we would call it a line segment.

A line segment has two end points, and a line continues in both directions. If you see 𝐴𝐵 with a line above it that does not have arrows, that is indicating a line segment. And it tells us that it does not extend infinitely in either direction. We should also note that any points that fall on the same line are called colinear points. And if you have a point that does not fall on the same line, then the two points are said to be noncolinear points.

The final object in space we want to consider is a plane. A plane is a flat surface made up of points that extend infinitely in all directions. When we’re sketching a plane, we usually use a quadrilateral to represent that. The key property we need to remember about a plane is that there is exactly one plane through any three noncolinear points. We also know that points or lines that fall on the same plane can be called coplanar, while points or lines that do not fall on the same plane would be called noncoplanar.

We can label a plane with any three of the noncolinear points on that plane, here, plane 𝐴𝐵𝐶. But we should also note that the order of the capital letters does not matter. We could call this plane plane 𝐵𝐴𝐶. You might also see a plane labeled with a capital script letter. In this case, we could call it plane 𝐾. Now, let’s consider the way that these points lines and planes can interact with each other in space.

When it comes to lines in space, we can categorize their relationships in two different ways, lines that intersect and lines that do not intersect. For any two intersecting lines in space, they can be found on the same plane. We can say this because for any pair of intersecting lines, we know that we’ll have at least three noncolinear points Here, 𝐴 is not found on the line through 𝐶𝐷. By showing that at least three noncolinear points are here, we can say they exist in the same plane.

What about lines that do not intersect? Almost immediately, you probably thought of parallel lines because we know that parallel lines do not intersect. Another key property of parallel lines is that they are the same distance from one another at every point. If we think about some points on parallel lines, we recognize that in a pair of parallel lines, there exist at least three noncolinear points. And therefore, for a set of parallel lines in space, they will fall on the same plane. We’ve just shown that in space parallel lines and intersecting lines are coplanar.

We’re used to having just two categories of lines, intersecting or parallel. However, in space, we have a third category. Lines that do not intersect and are noncoplanar are called skew lines. And here’s what a pair of skew lines might look like. These two lines would not intersect because the line ℎ occurs in a vertically higher plane than the line 𝑔. However, they do not fit the definition of parallel lines because at some points line 𝑔 is closer to line ℎ than at other points, making skew lines, lines that do not intersect and do not fall on the same plane.

Let’s take the properties we’ve just looked at and answer some questions about the intersections of points, lines, and planes.

How many planes can pass through three noncolinear points?

Imagine that we have three arbitrary points in space 𝐴, 𝐵, and 𝐶. We know that between any two points, there exists only one line, which means one line passes through the point 𝐴𝐵; one line could pass through the point 𝐴𝐶, which would create a set of intersecting lines. And if there wasn’t a line passing from the point 𝐴𝐶, if the line through 𝐶 is parallel to the line 𝐴𝐵, it is still true that 𝐴, 𝐵, and 𝐶 are noncolinear. They’re not on the same line. But parallel lines in space and intersecting lines in space are coplanar; they exist on the same plane. And this means that through any three non colinear points, there will be exactly one plane.

Based on the properties of points, lines, and planes in space, we can say that there exists exactly one plane through any three noncolinear points.

In our next example, we need to find the intersection between a line segment and a plane.

What is the intersection between segment 𝐵𝐵 prime and the plane 𝐴𝐵𝐶?

Let’s start by identifying the plane 𝐴𝐵𝐶. This will be the plane that contains the three noncolinear points, 𝐴, 𝐵, and 𝐶. It, of course, would include this triangular piece, but it also extends in all directions. Next, we can identify the segment 𝐵𝐵 prime, which is here. The line 𝐵𝐵 prime would extend in both directions, but we’re only interested in the segment between 𝐵 and 𝐵 prime.

When we look for the intersection, we’re looking for points that are common to both objects. And in this case, that’s only 𝐵. The plane 𝐴𝐵𝐶 and the segment 𝐵𝐵 prime only share one point, point 𝐵. You could maybe imagine this a bit like if you balanced a pencil on top of a flat sheet of paper, where the pencil represents a line segment and the piece of paper represents a plane. The intersection there would be a single point. And that single point in the case of this object would be point 𝐵.

In our next example, we’ll consider the intersection of two planes.

What is the intersection of the plane through 𝐴𝐵𝐵 prime 𝐴 prime and the plane through 𝐵𝐶𝐶 prime 𝐵 prime?

We’ve been given four points to identify our plane. We see that 𝐴𝐵𝐵 prime 𝐴 prime represents a face on this rectangular prism. However, the plane that passes through this point would be larger than this, as this plane extends in all directions. Next, we need to identify the plane that would pass through 𝐵𝐶𝐶 prime 𝐵 prime. Again, we have a plane that is passing through a face on a rectangular prism. But again, we know that these planes extend infinitely in all directions.

By extending that 𝐵𝐶𝐶 prime 𝐵 prime plane, we see that it is slicing through the 𝐴𝐵𝐵 prime 𝐴 prime plane, and the intersection is happening at the line that passes through the points 𝐵𝐵 prime. The intersection of these two planes form a line, and we can name this line 𝐵𝐵 prime.

In our next example, we’ll see what can happen at the intersection of three different planes.

What is the intersection of the planes 𝑀𝐴𝐵, 𝑀𝐵𝐶, and 𝑀𝐴𝐶?

First, let’s look at 𝑀𝐴𝐵. The plane 𝑀𝐴𝐵 would be the plane that contains this face in our triangular pyramid. If we do the same thing for the points 𝑀𝐵𝐶, it’s here. If we were just speaking of the intersection of the plane 𝑀𝐴𝐵 and 𝑀𝐵𝐶, that intersection is the line containing the points 𝑀 and 𝐵. If we add in this final plane 𝑀𝐴𝐶, this space here, we then end up with an intersection of the plane 𝑀𝐴𝐵 and 𝑀𝐴𝐶 at the line 𝑀𝐴. And we see the plane 𝑀𝐵𝐶 intersects the plane 𝑀𝐴𝐶 at the line 𝑀𝐶.

But now, we need to consider what is the shared space from all three of these planes. And the only shared space among all three of these planes is the point 𝑀. If we visualized this pyramid from the top down, again, we can see that the common space to all three of these planes is the point 𝑀.

Let’s consider one final example of the intersection of two planes.

𝑋 and 𝑌 are planes that intersect at line 𝐿. 𝐵 is a point on plane 𝑋, and 𝐶 is a point on plane 𝑌. Determine the intersection of plane 𝑌 with plane 𝐴𝐵𝐶.

In our figure, we can see line 𝐿 is the intersections of plane 𝑋 and 𝑌. We want the intersection of plane 𝑌 with plane 𝐴𝐵𝐶. Let’s first identify plane 𝑌. We can see that the plane 𝑌 includes the points 𝐴 and 𝐶. And if we’re interested in plane 𝐴𝐵𝐶, we recognize that the points 𝐴 and 𝐵 are located along the same plane, plane 𝑋, and the points 𝐶 and 𝐴 are both located in the plane 𝑌. This tells us that the point 𝐴 is located in plane 𝑋 and in plane 𝑌.

Because the points 𝐴, 𝐵, and 𝐶 are noncolinear, they can form exactly one plane. To see plane 𝐴𝐵𝐶, first, we’ll connect a line through points 𝐴 and 𝐵 and another line through the points 𝐴, 𝐶. From there, we can add an additional line that goes through point 𝐶. By connecting these lines, we have some idea of how this plane would look.

At this point, we should start to see that the plane 𝐴𝐵𝐶 is slicing through the plane 𝑌. And this slicing is happening along the line that goes through the points 𝐴 and 𝐶. Plane 𝑌 contains the line 𝐴𝐶, as does plane 𝐴𝐵𝐶, which makes the intersection of plane 𝑌 and plane 𝐴𝐵𝐶 the line 𝐴𝐶.

Before we finish, let’s quickly review our key takeaways from this video. For any two lines in space, the possible configurations will be parallel, intersecting, or skew. A plane can be defined by three noncolinear points or two intersecting lines. And finally, for a line and a plane in space, the possible configurations will be intersecting at a point, line included in the plane, or line parallel to the plane but not included.