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

In this explainer, we will learn how to identify and model geometric concepts like points, lines, and planes in space along with their properties.

You might have heard of a man called Euclid who is sometimes called the father of geometry. The word geometry comes from the Greek prefix β€œgeo-,” which means earth, and the suffix β€œ-metry,” which means the process of measuring. We understand the world around us from his writings on solid geometry, which deals with 3-dimensional objects, such as cubes and spheres, and plane geometry, which deals with objects in 2-dimensional space. In particular, points, lines, and planes are all geometric concepts that relate to positions in space and provide a starting point to defining all other geometrical concepts.

Fundamentally, we start with a point. A point is a location in space that has neither shape nor size. However, in order to be able to model this concept, we use a small dot to represent the idea of a point. A point is usually defined by a capital letter.

Definition: Point

A point is a location in space. It has neither shape nor size.

Given any two points in space, we can draw exactly one straight line between those points. It is particularly important to remember that when we work with lines in space, we are always working with straight lines that extend infinitely in both directions. We draw these as straight lines with arrows on either end. A line is usually named using two of the points on that line.

Three or more points that lie on the same line are called collinear points. If a point does not lie on the same line as those other points, we say that this set of points is noncollinear.

We should also note here that the distance between any two points on a line is called a line segment. We say the line that joins points 𝐴 and 𝐡 and terminates at each end is line segment 𝐴𝐡, or 𝐴𝐡.

Definition: A Straight Line

A line is a connected set of points that extends infinitely in two directions. A line passing through points 𝐴 and 𝐡 can be named in a number of ways. For instance, the line passing through 𝐴𝐡 can be defined by ⃖⃗𝐴𝐡, ⃖⃗𝐡𝐴, line 𝐴𝐡, line 𝐡𝐴, or line 𝑙.

The third concept we will consider is that of a plane. A plane is a 2-dimensional surface made up of points that extends infinitely in all directions. There exists exactly one plane through any three noncollinear points. Three or more points that lie on a plane are called coplanar. If a point does not lie on the same plane as the other three points, this set of points is called noncoplanar. We can also define a plane in terms of two parallel lines, two intersecting lines, or a line and an external point. This is because two parallel lines, two intersecting lines, and a line and a point will all have at least three noncollinear points.

A plane is usually defined using a single uppercase letter or, rarely, using three or more of the noncollinear points in that plane. You will usually see planes modeled as a quadrilateral. The plane shown can be defined as plane 𝐾, plane 𝐴𝐡𝐢, plane 𝐡𝐴𝐢, or plane 𝐢𝐡𝐴.

Definition: Planes

A plane is a 2-dimensional surface made up of points that extends infinitely in all directions. There exists exactly one plane through any three noncollinear points.

Of particular interest to us as we work with points, lines, and planes is how they interact with one another.

Recall that for any point in space, there exist an infinite number of lines passing through that point. This same principle is true for planes. For any point in space, there will be an infinite number of planes passing through that point.

In our first two examples, we will demonstrate how to identify a number of lines or planes passing through a point.

Example 1: Finding the Number of Straight Lines Passing through a Specific Point in Space

Find the straight lines that pass through the point 𝐡.

Answer

In this figure, we see a few different line segments that include point 𝐡.

𝐴𝐡, 𝐢𝐡, and 𝑀𝐡 are all line segments that have an endpoint at 𝐡. We know that there exist lines through any two points in space, which means there will be three lines through point 𝐡 that we can label: ⃖⃗𝐴𝐡,⃖⃗𝐢𝐡,⃖⃗𝑀𝐡.and

The straight lines that pass through point 𝐡 are ⃖⃗𝐴𝐡, ⃖⃗𝐢𝐡, and ⃖⃗𝑀𝐡.

Example 2: Identifying the Planes that Pass through Specific Points

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

Answer

The planes that pass through both points 𝐴 and 𝐡 will be the planes that pass through the line ⃖⃗𝐴𝐡. We know that a plane can exist through any three noncollinear points. In this rectangular prism, we can visualize these planes as the space that will contain certain faces of the prism. One of the faces will be the face that contains the two parallel lines ⃖⃗𝐴′𝐡′ and ⃖⃗𝐴𝐡. We can call that plane 𝐴𝐡𝐡′.

Another plane exists that contains the two parallel lines ⃖⃗𝐴𝐡 and ⃖⃗𝐢𝐷. Let’s call this plane 𝐴𝐡𝐢.

The third plane is not immediately obvious. We recall that any three noncollinear points form a plane. Therefore, there exists some plane that will contain points 𝐢′, 𝐴, and 𝐡. This third plane is 𝐴𝐡𝐢′.

Three planes that pass through points 𝐴 and 𝐡 are 𝐴𝐡𝐢, 𝐴𝐡𝐡′, and 𝐴𝐡𝐢′.

The Relationship between Two Lines in Space:

There are three possible relationships that two lines can have. First, consider a pair of coplanar lines. These lines might intersect at any angle, as demonstrated in the following diagram, or they could be perpendicular (i.e., they intersect orthogonally).

If coplanar lines do not intersect, then they are parallel. They will never meet.

A pair of lines that neither intersect nor are parallel to one another are said to be skew. Skew lines can only exist in 3 dimensions.

The Relationship between a Line and Plane in Space:

There are three relationships that a line and a plane can have.

A line might lie on a plane. In this case, every point on the line will lie on the plane.

If a line intersects a plane, the intersection means sharing a common point that lies on both of them.

A line can also intersect a plane orthogonally, in which case the line is said to be perpendicular to the plane. Then, this line is perpendicular to all lines on that plane that intersect this line.

If a line does not intersect a plane, the line is parallel to the plane.

The Relationship between Two Planes in Space:

Finally, there are three possible relationships that can exist between two planes in space. If the two planes share all points, they are said to be coincident.

If two planes intersect, the intersection is always a line. These two planes might intersect orthogonally, so they are said to be perpendicular.

These two planes do not intersect. They are parallel planes.

Note

While we have been considering a pair of planes in space, three planes intersecting can share a common point.

In the next example, we will demonstrate how to identify relationships between line segments in a rectangular prism.

Example 3: Identifying the Relation between Line Segments in Space

Consider the rectangular prism 𝐴𝐡𝐢𝐷𝐸𝐹𝐺𝐻, where 𝐴𝐡≠𝐡𝐢≠𝐴𝐸.

  1. What can be said about 𝐸𝐹 and 𝐹𝐺?
    1. They are parallel.
    2. They are perpendicular.
    3. They are neither parallel nor perpendicular.
    4. They are skew.
  2. What can be said about 𝐴𝐸 and 𝐢𝐺?
    1. They are skew.
    2. They are perpendicular.
    3. They are parallel.
    4. They are neither parallel nor perpendicular.
  3. What can be said about 𝐡𝐷 and 𝐷𝐻?
    1. They are perpendicular.
    2. They are neither parallel nor perpendicular.
    3. They are skew.
    4. They are parallel.
  4. What can be said about 𝐡𝐷 and 𝐴𝐢?
    1. They are parallel.
    2. They are perpendicular.
    3. They are neither parallel nor perpendicular.
    4. They are skew.

Answer

Part 1

In the rectangular prism, we want to identify the relationship between different pairs of line segments, specifically 𝐸𝐹 and 𝐹𝐺.

Rectangular prisms are made up of 6 rectangular faces, and in a rectangle, adjacent sides are perpendicular. Since 𝐸𝐹𝐺𝐻 is a rectangle, we can therefore say that 𝐸𝐹 and 𝐹𝐺 meet at an angle of 90∘. So, the answer is option B; 𝐸𝐹 and 𝐹𝐺 are perpendicular.

Part 2

𝐴𝐸 and 𝐢𝐺 are line segments that lie on opposite faces of the rectangular prism.

They do not intersect. Since these two faces are opposite faces in a rectangular prism, we can say that 𝐴𝐸 and 𝐢𝐺 are parallel. The answer is option C.

Part 3

𝐡𝐷 and 𝐷𝐻 are line segments that occur on perpendicular faces of the prism and intersect at point 𝐷.

This means that at the point of intersection, 𝐡𝐷 and 𝐷𝐻 are perpendicular. This is option A.

Part 4

𝐡𝐷 and 𝐴𝐢 are line segments that lie on the same plane, 𝐴𝐡𝐢. Since the line segments are not parallel, they must intersect.

We know that the diagonals of a rectangle are not perpendicular, so 𝐡𝐷 and 𝐴𝐢 are neither parallel nor perpendicular. The answer is option C.

Note

Although ⃖⃗𝐡𝐷 and ⃖⃗𝐴𝐢 are neither parallel nor perpendicular, this does not mean that they are skew lines. ⃖⃗𝐡𝐷 and ⃖⃗𝐴𝐢 are not skew lines since they intersect and lie on the same plane.

We will now demonstrate how we can identify skew lines from an image.

Example 4: Identifying Skew Lines

Using the rectangular prism below, decide which of the following is skew to ⃖⃗𝐢𝐺.

  1. ⃖⃗𝐢𝐡
  2. ⃖⃗𝐷𝐢
  3. ⃖⃗𝐸𝐻
  4. ⃖⃗𝐹𝐡
  5. ⃖⃗𝐻𝐺

Answer

Recall that skew lines are lines that do not intersect but are not parallel. Skew lines are noncoplanar and therefore can only exist in 3 dimensions. A line that is skew to ⃖⃗𝐢𝐺 cannot be parallel to ⃖⃗𝐢𝐺, nor can it intersect that line. This means the only lines that can be skew to ⃖⃗𝐢𝐺 are ⃖⃗𝐸𝐻, ⃖⃗𝐴𝐷, ⃖⃗𝐴𝐡, and ⃖⃗𝐸𝐹.

From the list of options, the correct answer is C. ⃖⃗𝐸𝐻 is skew to ⃖⃗𝐢𝐺.

The figure in the next example will show us a possible configuration of a line and a plane.

Example 5: Identifying the Intersection between a Plane and a Line

Observe the given figure and choose the correct statement.

  1. The straight line is parallel to the plane.
  2. The straight line is contained within the plane.
  3. The straight line intersects the plane.

Answer

The figure shows a plane, defined as 𝑋, that extends infinitely in all directions. We observe from the diagram that point 𝐴 lies on plane 𝑋.

We also see that point 𝐴 lies on line 𝐿. Since plane 𝑋 and line 𝐿 share a common point, that is, point 𝐴, we can say that the straight line intersects the plane. Therefore, option C is correct. The straight line intersects the plane.

The next example is a possible configuration of two planes in space.

Example 6: Identifying the Intersection between Two Planes

What is the intersection of the plane through 𝐴𝐡𝐡′𝐴′ and the plane through 𝐡𝐢𝐢′𝐡′?

Answer

In this rectangular prism, the plane that contains the points 𝐴, 𝐡, 𝐡′, and 𝐴′ is the plane that contains the vertical face on the right of the diagram. The plane that contains the points 𝐡, 𝐢, 𝐢′, and 𝐡′ is the plane that contains the vertical face at the front of this prism. The intersection of these two faces is a line. We know this is true because both of these faces contain the points 𝐡 and 𝐡′. The line between 𝐡 and 𝐡′ will be the line of intersection of these two planes.

We see highlighted here that the shared line and hence the intersection of plane 𝐴𝐡𝐡′ and plane 𝐡𝐢𝐡′ is ⃖⃗𝐡𝐡′.

The final example shows us a configuration of three planes in space.

Example 7: Identifying the Intersection between Three Planes

What is the intersection of planes 𝑀𝐴𝐡, 𝑀𝐡𝐢, and 𝑀𝐴𝐢?

Answer

This pyramid is made up of four triangular faces. Since there exists exactly one plane through any three noncollinear points, each of the triangular faces lies on a unique plane. We are interested in the planes 𝑀𝐴𝐡, 𝑀𝐡𝐢, and 𝑀𝐴𝐢, each of which contains the point 𝑀. This means that point 𝑀 lies on all three of these planes. Since point 𝑀 is a shared point on each of these planes, the intersection of these planes will be {𝑀}.

Let us finish by recapping the key points.

Key Points

  • A point is a location in space. It has neither shape nor size.
  • A line is a connected set of points that extends infinitely in two directions.
  • A plane can be defined by three noncollinear points, two parallel lines, or two intersecting lines.
  • A set of points are said to be collinear if they lie on the same line. If not, they are said to be noncollinear.
  • A set of points are said to be 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 angle, or perpendicular.
  • For any two lines in space, the possible relationships will be parallel, intersecting with angle, perpendicular, or skew.
  • For a line and a plane in space, the possible configurations will be intersecting at a point (with any angle), perpendicular, included in the plane, or parallel to the plane.
  • For any two planes, the possible configurations will be coincident, parallel, intersecting at a straight line (with any angle), or perpendicular. Three planes can intersect at one point, not a straight line.

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