Lesson Video: Points, Lines, and Planes in Space Mathematics

Video Transcript

In this video, we will learn how to identify and model points, lines, and planes in space. We’ll also consider their properties, along with how they interact with one another. So let’s begin by understanding exactly how we can define these three objects.

In geometry, a point is a location in space. 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 point, and generally it is given a letter as a name. We can then go further and say that we can take any two points in space. We can join these two points with exactly one straight line between these points. This allows us to define a line as a connected set of points that extends infinitely in two directions.

We can define this line that joins 𝐴 and 𝐵 in a number of ways, for example, line 𝐴𝐵 with a double-ended arrow over the letters, line 𝐵𝐴, or using the words “line 𝐴𝐵” or “line 𝐵𝐴,” or even by defining the line with a single letter, such as line 𝐿. Note that the distance between any two points on a line is called a line segment. This then leads us to the third important definition for this video, planes.

A plane is a two-dimensional surface that extends infinitely in all directions. We usually model a plane like this. And the defining property of a plane also helps us with naming planes. There exists exactly one plane through any three noncollinear points. That means that if points 𝐴, 𝐵, and 𝐶 lie on a plane, we can define the plane as plane 𝐴𝐵𝐶 or even plane 𝐵𝐴𝐶 or plane 𝐶𝐵𝐴. Or just like we did with a line, we can define the plane with a single letter, for example, plane 𝐾.

We’ll now see an example where we identify and define the planes which pass through two given points.

Find three planes that pass through both of the points 𝐴 and 𝐵.

Let’s observe that points 𝐴 and 𝐵 lie on the base of this diagram. The planes that pass through both points 𝐴 and 𝐵 will be the planes that pass through the line 𝐴𝐵. We should recall that there exists exactly one plane through any three noncollinear points. Notice how we use the word “noncollinear.” If the three points were collinear, they would lie on a line. And there would be an infinite number of planes passing through three collinear points.

So, in order to find a plane passing through 𝐴 and 𝐵, we need to find another point which doesn’t lie on this line, which could define a plane. Let’s take point 𝐶 and visualize the plane that would pass through 𝐴, 𝐵, and 𝐶. It would look something like this. And so one of the planes we are looking for is actually one which contains one of the faces of this prism.

One way in which we can define a plane is by using three points that we know lie on the plane. So we could call this plane the plane 𝐴𝐵𝐶 or plane 𝐴𝐶𝐵. However, the plane 𝐴𝐵𝐷 would also work.

Now let’s see if we can find another plane that passes through 𝐴 and 𝐵. We can do this by finding another point. So let’s use the point 𝐵 prime. Since we know that 𝐵 prime is not collinear with 𝐴 and 𝐵, then we can create another plane. We can define this plane as the plane 𝐴𝐵𝐵 prime. We know that we are looking for three planes passing through 𝐴 and 𝐵. So let’s see if we can find the third plane.

Now, it might be tempting to think that as we found a plane on the base of the prism and one on the side of the prism that perhaps there is another plane passing through 𝐴 and 𝐵 on one of the other sides of the prism. Perhaps there might be one on the back of the prism. However, if we visualize a plane here, it contains the points 𝐴, 𝐴 prime, 𝐷, and 𝐷 prime, but it does not contain the point 𝐵. So that wouldn’t work.

In fact, the third plane containing points 𝐴 and 𝐵 will look something like this. This plane does contain the points 𝐴 and 𝐵 and 𝐶 prime and 𝐷 prime. So we could define it as the plane 𝐴𝐵𝐶 prime, although it would be equally valid to refer to it as plane 𝐴𝐵𝐷 prime. Therefore, we can give the answer that the three planes passing through the points 𝐴 and 𝐵 are the planes 𝐴𝐵𝐶, 𝐴𝐵𝐵 prime, and 𝐴𝐵𝐶 prime.

So far, we have covered exactly what we mean by points, lines, and planes in space. Now, let’s think about the way that these interact with each other in space.

When it comes to two lines in space, there are three possible relationships that these lines can have, which can be determined by whether the lines intersect or do not intersect.

Firstly, let’s think about two lines which are coplanar, which means that they lie on the same plane. Here are two lines which lie on the same plane, and they are intersecting. Furthermore, if they intersect at right angles, we could say that the lines are intersecting orthogonally. We can also have two lines which are coplanar and are nonintersecting, in which case these lines would have to be parallel. They will never meet.

Now, we’re used to having two types of relationships between lines: those that intersect and those which don’t. However, in space, there is another option. These types of lines will be noncoplanar and nonintersecting. They can be modeled like this. These two lines, ℎ and 𝑔, are called skew lines. They are noncoplanar lines that don’t intersect because line ℎ occurs in a vertically higher plane than line 𝑔. They also cannot be said to be parallel, because at some points line 𝑔 is closer to line ℎ than at other points. Notice that skew lines can only exist in three dimensions.

So, now we have considered how two lines interact in space, let’s consider how a line and a plane might interact. Once again, we have three different possible ways that a line and a plane can interact in space. The first way is that every point on the line also lies on the plane. The second type of relationship that a line and a plane can have is that they are intersecting, in which case the intersection point is a shared, common point that lies on both of them. If the line is perpendicular to the plane, then we can say that the line and the plane intersect orthogonally.

This brings us to the last type of interaction, when the line and plane do not intersect. But of course because the line and plane extend infinitely in both directions, then there is only one way in which a line and a plane cannot be intersecting. And that is if the line and plane are parallel.

We’ll now see the final possible set of interactions, which will be between two planes in space. Just as we saw in the previous geometrical interactions, there are three possible relationships between two planes. Firstly, if the two planes share all points, then we say that they are coincident. Secondly, we can have intersecting planes. We could model intersecting planes like this. As the lower diagram shows, we can also have two planes which intersect orthogonally.

Notice that when two planes intersect, the intersection is always a line. And of course we can have planes that are nonintersecting. If two planes do not intersect, they must be parallel. As a final note on intersecting planes, here we have just been considering the intersection of two planes. But if three planes intersect, then they can share a common point.

We’ll now see an example where we need to identify the relationship between a given line and plane.

Observe the given figure and choose the correct statement. Option (A) the straight line is parallel to the plane. Option (B) the straight line is contained within the plane. Or option (C) the straight line intersects the plane.

In the figure, we can observe that we have a plane, defined as 𝑋, and a line, defined with the letter 𝐿. The line and the plane will extend infinitely in all directions. We can also see that there is a point 𝐴 which lies on the plane. Notice that the point 𝐴 also lies on the line. Then, because the plane and the line have a common point 𝐴, we can say that the straight line 𝐿 intersects the plane. Therefore, the correct statement is that given in option (C). The straight line intersects the plane.

But before we finish this question, it might be worth considering what the other two answer options would look like on a diagram. Option (A) gives a statement about a line which is parallel to a plane. A line which is parallel to a plane can be modeled as something like this. Recall that a line and a plane that are parallel will never intersect. But we know that in the diagram we were given, the line and the plane intersect at point 𝐴. So the statement given in answer option (A) is incorrect.

In option (B), we are considering the statement that the line is contained within the plane. A line which is contained within a plane might look something like this. When a line is contained within a plane, every point on the line also lies on the plane. And it is at this point that we can note that the way in which the original diagram was drawn is very important. The dashed section of line 𝐿 indicates that not all of the points on the line are contained on the plane 𝑋. Therefore, the correct statement is that the straight line intersects the plane.

We can now summarize the key points of this video. We began by defining that a point is a location in space. It has neither shape nor size. Then, a line is a connected set of points that extends infinitely in two directions. A plane can be defined by three noncollinear points or by two parallel lines or two intersecting lines. We say that a set of points are coplanar if they lie on the same plane. If not, they are said to be noncoplanar.

For any two coplanar lines, the possible relationships are parallel, intersecting with an angle, or perpendicular. For any two lines in space, the possible relationships are parallel, intersecting with an angle, perpendicular, or skew. For a line and plane in space, the relationships can be intersecting with any angle, perpendicular, included in the plane, or parallel to the plane.

Finally, we saw that two planes can have the following configurations. They can be coincident, parallel, intersecting at a straight line with any angle, or perpendicular. But if we are considering three planes and not just two, they can intersect at one point, not a straight line.