Point on a Line, a Line on a Plane, and a Plane in Space

Each point has a specific location. Two points determine a line. Three points determine a plane. Let us consider some simple math derivations to arrive at a format for each. For simplicity’s sake, we will use the familiar x, y, z Cartesian coordinate system. We begin with a point on a line.

First, Point on a Line

In space, a single point has an x value, a y value, and a z value. If the coordinate system chosen for the point is a simple 1-D line, then only one variable – say x – is needed to describe it. Then, since there is no y or z to consider, the mathematical description of the point is

x = c

But let us, for reasons that will be understood later, write it using two constants instead of one

Ax = C [Equation of a Point]

Second, Line in a Plane

We move on from the point on a line. Choosing the xy-plane, we will choose two distinct points (x₁,y₁) and (x₂,y₂).¹ The slope of the line, m, is

m = (y₂– y₁) / (x₂-x₁)

Since a line is infinitely long, every line must either intersect the x or the y-axis, and most often, both. For derivation purpose, we can choose whichever axis is intersected as our y-axis. The intercept point we label (0,b). We choose that point to represent (x₁,y₁). This results in

m = (y₂-b) / (x₂-0)

Removing the subscript and multiplying through,

y = mx + b

Is there a superior general equation for a line? Yes.

Ax + By = C [Equation for a Line]

This equation appeals to the intellect. Consider: Move the expression in x and the constant, c, to the right-hand side of the equation. Divide both sides by b, and you get?

y = (-A/B)x + C/B

This is the same style format as the slope-intercept equation for a line. The -a/b is equivalent to the slope, m, while the far right term is equivalent to the intercept.

Now we have discussed only lines within the xy-plane. But the letters x and y were chosen at random. Any other letters, say v and w would do as well.

Third, Plane in Space

Extending the logic above, we now have the three points, (x₁,y₁,z₁), (x₂,y₂,z₂), and (x₃,y₃,z₃). From these three points, three combinations of two points can be drawn from “the hat.” Through each of the pairs a single line can be drawn.² However, it takes only two lines with two “slopes” to characterize a plane.

The general equation of a line in a plane was given above. Logically, and by extension, the equation for a plane in space is,

Ax + By +Cz = D [Equation of a Plane]

Consider the example graph for z = 2x + 4y, below.

In Conclusion

What an interesting thing it is to compare the mathematical expressions next to each other, for a point in a line, a line in a plane, and a plane in a coordinate system.

• Ax = C [point]
• Ax + By = C [line]
• Ax + By + Cz = D [plane]

What comes after that? Whatever it is (let’s call it t) ought to be described by

Ax + By + Cz + Dt = E

¹ Three distinct points for 3D (all 3 cannot satisfy the same y = mx + b equation).
² Any pair of lines from the 3 points will intersect.

Thanks to mathematician Mike DeHaan for proofreading the material.

Note: You might also enjoy Mathematical Equation for a Cone

References:

• Ask Dr. Math: Equation of a Line in 3-D Space
• U. of Washington: Dept. of Mathematics: Basic Equations of Lines and Planes

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