Lines, planes and their equations

In this topic, we are going to study several geometry related concepts. We are going to talk about lines and planes, or, to be more exact, about their equations. This knowledge will be required later for solving linear systems.

Line

First thing we will examine is the line. We will talk about lines on the two-dimensional plane. Any line can be defined by an equation. It means that a line consists of all points with coordinates that satisfy this equation. There are a lot of types of such equations: general equations, line equations with the slope, normalized line equations, parametric line equations, line equations from two points and many more. In this topic we will only study two types that will prove themselves useful in the future for solving linear systems: general line equations and line equations from two points.

A general line equation has the following form: $ax+by+c=0$. $x$ and $y$ are unknowns, and $a$, $b$ and $c$ are coefficients, plus $a$ and $b$ can't be null at the same time. If that happens, the equation becomes identical to $c=0$, and that doesn't define a line.

Moreover, it is worth noting that if $a=0$ then the line will be parallel to the $Ox$ axis, and if $b=0$, then the line will be parallel to the $Oy$ axis. If $c=0$ then the line passes through the origin of coordinates.

It's time for an example! Assume we have the following line equation: $2x-5y+5=0$. Take a look at the graph:

The line passes through two dots on the axes, $(0,1)$ and $(-2.5, 0)$. You can verify yourself that putting those values in the equation will result in it being correct.

But how, having the graph of the line, do we make the line equation from it? For that purpose there is the line equation from two points. It has the following form:

$$\frac{y-y_1}{y_2-y_1}=\frac{x-x_1}{x_2-x_1}$$Having two dots with coordinates $(x_1, y_1)$ and $(x_2,y_2)$, after substituting them in the equation we will get the line equation. Also the equation can be slightly rewritten to be more similar to the general line equation after substitution:

$$(y_1-y_2)x+(x_2-x_1)y+(x_1y_2-x_2y_1)=0$$Let's check if it is correct: from the previous graph we see that the line passes through two dots: $(0, 1)$ and $(-2.5, 0)$. We could choose any dots, but we are going to take these, since they will be easier to count. Substitute the two dots in the equation and see what is the result:

$(1-0)x+(-2.5-0)y+(0\cdot0-(-2.5\cdot1))=0$

$x-2.5y+2.5=0$

If we multiply the equation by two:

$2x-5y+5=0$

As you can see, we've got the same line equation.

Plane

Now it's time to talk about planes in a three-dimensional space. Just like a line, they can be defined in many ways, but we are interested only in two: the general plane equation and the plane equation from three dots which do not lie on the same line. Let's get to it!

The general plane equation looks this way: $ax+by+cz+d=0$.

You could notice that it is rather similar to the line equation, just with an extra unknown and a coefficient.

In order to make plane equation from three dots you need to use the following equation:

If you substitute the dots' coordinates to this equation and find the determinant of the matrix, you'll get the equation of the plane passing through these three dots. It is a rather time-consuming task and we won't need this in the future, but don't forget that you can do it!

If we were to consider a space with more than three dimensions, then the general plane equation will just get some extra unknowns and coefficients. For example, in five-dimensional space: $ax+by+cz+dw+ev+f=0$, where $a, b, c, d, e$ and $f$ are coefficients and $x, y, z, w$ and $v$ are the unknowns.

Conclusion

Further down the line all you need to remember are the general equations for a line and a plane. They have the following form:

line equation:

$$ax+by+c=0$$plane equation:

$$ax+by+cz+d=0$$

You will need this in the next topic where we will try to uncover the geometric interpretation of linear system solutions.