Points, Lines and Planes

Points describe locations without size

A point has position but no length, width or thickness. We name points with capital letters such as $A$, $B$ and $P$. The dot used to display a point must have visible size, but that size does not belong to the point itself.

Coordinates can specify the location. On a number line, one coordinate is enough. In a coordinate plane, an ordered pair such as $(2,3)$ locates a point. In three-dimensional space, an ordered triple such as $(2,3,4)$ is needed after a coordinate system has been chosen.

The same physical location can have different coordinates in different reference frames. Coordinates are a description relative to an origin, axes and units; they are not an intrinsic label attached to the location.

In an axiomatic treatment, points, lines and planes are usually primitive terms: their relationships are set out through assumptions rather than defined entirely using simpler geometric objects. David Joyce’s commentary on Euclid’s first definition explains this role. The descriptions here give useful intuition without claiming that a dot is a formal definition.

A line, a segment and a ray are different sets of points

A line is straight, has no thickness and extends without endpoints in both directions. Two distinct points determine exactly one line. The notation $\overleftrightarrow{AB}$ names the line through $A$ and $B$; a lowercase letter such as $\ell$ can also name it.

A segment, $\overline{AB}$, contains $A$, $B$ and the points between them. It has finite length, written $AB$. A ray, $\overrightarrow{AB}$, contains its endpoint $A$ and continues through $B$ indefinitely in one direction.

Reversing the naming points leaves a line or undirected segment unchanged. It changes the ray: $\overrightarrow{AB}$ starts at $A$, whereas $\overrightarrow{BA}$ starts at $B$. Arrowheads are therefore mathematical information, not just graphic styling.

A line contains segments, but it does not have the length of a displayed segment. Extending the drawing extends the visible portion; the line was already infinite.

Collinear points and segment addition

Points are collinear when one line contains them all. Two distinct points are always collinear because they determine a line. The condition becomes informative when three or more points are involved.

If $A$, $B$ and $C$ are collinear and $B$ lies between $A$ and $C$, then

Both conditions matter. Without betweenness, adding those lengths might traverse part of the line twice. If points have coordinates $-2$, 3 and 7 on a number line, in that order, the lengths are 5, 4 and 9 units, and $5+4=9$.

The midpoint of a segment divides it into two equal lengths. On a number line, the midpoint coordinate is the average of the endpoint coordinates. For endpoints $-2$ and 7, the midpoint is $(-2+7)/2=2.5$. It is 4.5 units from either endpoint.

A statement that a point is a midpoint therefore combines membership and measurement: the point lies on the segment and the two resulting segments have equal lengths.

Planes extend in two independent directions

A plane is a flat two-dimensional surface that extends indefinitely. It has no thickness and no boundary. A slanted parallelogram is often used to draw a portion of a plane in a perspective view, but the parallelogram’s edges are not edges of the plane.

A plane may be named with a Greek letter, such as $\alpha$, or by three non-collinear points it contains, such as plane $ABC$. Three non-collinear points determine exactly one plane. Three points on the same line do not: infinitely many planes can pass through that line.

To understand why, imagine a plane rotating around a line through two fixed points. A third point on that same line cannot select one orientation. A third point off the line fixes the plane containing all three. The plane reference gives an algebraic description of the same object.

A plane does not have to be horizontal. A wall can model a vertical portion of a plane, and a tilted flat sheet can model an oblique portion. Flatness, not orientation, is the defining idea.

Coplanar does not mean collinear

Objects are coplanar when one plane contains them all. Any three points are coplanar. If they are non-collinear, their common plane is unique; if they are collinear, there are infinitely many choices.

Four points need not be coplanar. Three corners of a triangle on a table and one point above the table cannot all lie in the table’s plane. If the three corners are non-collinear, no different plane can contain all four either.

Collinearity is the more restrictive relationship. Collinear points are coplanar, but coplanar points can be spread throughout a plane. Similarly, if two points of a line lie in a plane, the whole line lies in that plane. Merely showing two projected dots inside the shaded patch of a picture is not enough to establish the original spatial positions.

How two lines can relate

Two distinct lines in a plane either intersect at one point or are parallel. They cannot intersect at two different points, because those two points determine a unique line. Lines that share every point are coincident: they are the same line described in different ways.

In three-dimensional space, a third possibility appears. Skew lines are non-coplanar lines. They do not intersect and are not parallel. MathWorld’s skew-line reference makes the distinction explicit.

A precise example uses coordinates. Let line $\ell$ consist of points $(t,0,0)$, where $t$ can be any real number. It is the $x$-axis. Let line $m$ consist of points $(0,s,1)$, where $s$ is any real number. It runs in the $y$ direction one unit above the $xy$-plane.

The lines have different directions, so they are not parallel. They cannot meet because every point on $\ell$ has $z=0$ and every point on $m$ has $z=1$. They are skew. Their projections can cross in a flat drawing without producing a shared point in space.

Perpendicular lines intersect at a right angle. The directions of the skew lines in this example are orthogonal, but the lines do not intersect. Under the geometric convention used here, we do not call them perpendicular lines. This keeps directional information separate from intersection information.

How a line and a plane can relate

A line may lie entirely in a plane, meet it at exactly one point, or have no point in common with it. In the last case it is parallel to the plane. We use “parallel to the plane” here for the disjoint case and describe containment separately.

Consider the plane $z=0$. The line $(t,0,0)$ lies in it. The line $(0,0,t)$ intersects it at the origin. The line $(t,0,2)$ lies two units above it and does not meet it.

A line is perpendicular to a plane when it meets the plane and is perpendicular to every line in the plane through that intersection point. A useful test is perpendicularity to two distinct intersecting lines in the plane at that point. Perpendicularity to just one line in the plane is insufficient.

For example, the $z$-axis is perpendicular to the $xy$-plane. A line in the $xy$-plane perpendicular to the $x$-axis is not therefore perpendicular to the whole plane; it may simply be the $y$-axis.

How planes can relate

Two distinct planes in three-dimensional Euclidean space either do not intersect or intersect along a line. If they do not meet, they are parallel. If they share every point, they are coincident descriptions of the same plane.

The coordinate planes provide concrete examples. The planes $z=0$ and $y=0$ meet along the $x$-axis, whose points are $(x,0,0)$. The planes $z=0$ and $z=2$ are parallel and disjoint.

Two distinct planes cannot intersect at only one point. An apparent corner shared by two shaded patches in a drawing does not limit the infinite planes to that corner. Three planes, however, can share exactly one point: $x=0$, $y=0$ and $z=0$ meet at the origin. Other arrangements of three planes may share a line or have no common point.

One equation can describe different objects

The surrounding space matters. On a number line, $x=2$ specifies one point. In a two-dimensional coordinate plane, it specifies a vertical line because $y$ remains free. In three-dimensional space, it specifies a plane because both $y$ and $z$ remain free.

Likewise, in three dimensions the pair of conditions $x=2$ and $y=-1$ describes a line:

Adding $z=4$ selects the single point $(2,-1,4)$. Each independent condition removes one free direction in this example. It is the set of allowed points, not the visual length of an equation, that determines the geometric object.

The 3D coordinate article develops these ideas with coordinate-plane diagrams, projections and distance calculations.

Read a spatial diagram carefully

Treat solid lines, dashed lines, point markers and plane labels as a notation system. Dashed lines may show hidden edges or construction guides; the caption should state which. A shaded plane patch shows orientation, not a finite boundary. A line drawn through another on the page may be in front of it or behind it.

Base a conclusion on the stated relationships or coordinates. When a diagram is schematic, do not infer equal lengths, perpendicularity or coplanarity from appearance alone. A useful explanation names the objects and the reason: “These planes intersect along the line where $y=0$ and $z=0$,” rather than “They meet along the edge in the picture.”

Related topics

Use Lines and Angles for the relationships formed by intersecting lines. Continue with 3D Coordinate Axes and Planes to locate and compare points in a fixed frame.