Equations of lines

There are many ways of representing a mathematical line. You are probably most familiar with point-y-intercept form,

$y=mx+b\text{.}$

That equation is useful when you know the slope and the y-intercept. Also, that form is easy to express in function notation, like $f(x)=mx+b\text{,}$ and it fits nicely with the other families of functions. Another way of describing a line is with standard form:

$ax+by+c=0\text{.}$

This form gives you both the intercepts, but most of the time it is inconvenient. Yet another type of line equation is point-slope form:

$y-y_0=m(x-x_0)\text{.}$

Not surprisingly, this is useful when you know a point and the slope.

Those three equations have something in common: they relate x to y. If you solve for y (already done for you in point-y-intercept form), you can choose an x-value and the equation will tell you the y-value. This isn’t the only way of doing it, as we will see. In particular, it doesn’t scale well to three-space; our new method will.

We are going to explore another triplet of line equations, starting with the vector equation of the line. To describe a line with vectors, we need an initial point that falls on the line—we’ll call it point A. From point A, we need a vector that tells us which way the line goes—we’ll call this $\overrightarrow{m}\text{.}$ Here’s what this might look like:

To get from point A to some point P elsewhere on the line, all we have to do is add a scalar multiple of the direction vector:

$\overrightarrow{OP}=\overrightarrow{OA}+t\overrightarrow{m}\text{.}$

The value of t controls how far along the line we go from A. If it is negative, we go in the opposite direction. Instead of using origin-to-point notation, we usually use $\overrightarrow{r}\text{,}$ which is implied to be a position vector (the r stands for radius, and a position vector is a radius from the origin). Here is the conventional way of writing the vector equation of the line:

$\overrightarrow{r}={\overrightarrow{r}}_0+t\overrightarrow{m}\text{,}$ $t\in ℝ\text{.}$

Suppose we substitute for vectors in ${ℝ}^2$ with variables as components:

$[x,y]=[x_0,y_0]+t[a,b]\text{,}$ $t\in ℝ\text{.}$

Breaking this up into two equations gives us the parametric equations of a line (two equations because there are two components):

$x=x_0+ta$ and $y=y_0+tb\text{,}$ $t\in ℝ\text{.}$

There is an important difference between this form and the earlier ones. The other equations relate x and y such that we can choose one an obtain the other. With the parametric equations, we choose a value for the free parameter t. We plug this into the equations and they tell us x and y. Instead of x being related to y, they are both independently controlled by t.

If we solve the parametric equations for t, we get the symmetric form:

$t=\frac{x-x_0}{a}=\frac{y-y_0}{b}\text{.}$

This form makes it easy to test if a particular point falls on the line: just evaluate both sides and see if they produce the same t-value.