Line

A line is the second simplest geometric object, and it is one-dimensional (it has only length, so to speak). Put simply, a line is an infinitely long straight line that has neither an end nor a beginning.

Basic properties

A line is usually named with a lowercase letter, for example a. A line is usually given by two points, because exactly one line passes through any two points. There is also a ray (or half-line), which is similar to a line except that it has a starting point (but still no end). For example, the arms of an angle are rays.

Describing a line in the plane

In the plane we can describe a line using a linear function, whose graph is always a line. Unfortunately this method fails in space, because a linear function cannot give you a third dimension. In the plane, though, a linear function is the simplest way to describe any line that is not parallel to the y-axis. If you are given the function, you can easily draw the line, but going the other way can be a problem. So let’s take a line that has already been drawn:

The rule for a linear function looks like this: y = mx + b, where m is the slope (gradient) of the line and b is the y-intercept. First we find b. The easiest way is to read off the graph where the line crosses the y-axis, that is, where x is zero. We can see that it is two, so b = 2. This is because when mx is zero, the only way to get y equal to two in y = mx + b is for b to equal two.

Now we have to work out what m equals. It will be easier if we temporarily remove the y-intercept and assume it is zero. The line then moves to the origin of the coordinate system, and we can find m more easily:

We can clearly see that at x = 6, y has the value 2. We substitute this into the rule for the function: 6m + 0 = 2. From this we easily work out that $m=\frac{2}{6}=\frac{1}{3}$. Now that we know the value of m, we can write the whole rule for the function, and so for the line: y = x/3 + 2.

$$y=\frac{x}{3}+2$$

Relative positions of lines

Two lines in the plane can be in several different positions relative to each other. Let’s start with positions in the plane.

If you have two lines that lie on top of each other and merge into one – they meet at every point – they are called coincident lines. If two lines meet at exactly one point, they are called intersecting lines. If two lines do not meet at any point, they are called parallel lines. Parallel lines are usually marked with two short dashes on each line, as in the last figure. The following figures sum it up:

You can find out how to determine the relative position of two lines in a separate article in the section on coordinate geometry. The same category also covers how to find the general equation of a line, the parametric equations of a line and the slope-intercept form of a line.