*The purpose of this post is to briefly discuss parallelism and perpendicularity of lines in both a geometric and algebraic setting.*

## Lines

What is a line? In Euclidean Geometry we usually don’t define a line and instead call it a primitive object (*the properties of lines are then determined by the axioms which refer to them*). If instead points and line *segments* – defined by pairs of points –* * are taken as the primitive objects, the following might define lines:

**Geometric Definition Candidate**

A *line, *, is a set of points with the property that for each pair of points in the line, ,

.

In terms of a picture this just says that when you have a line, that if you take two points *in *the line (the language *in *comes from set theory), that the line segment is a subset of the line:

### Exercise:

*Why is this *objectively *not a good definition of a line.*

Once we move into Cartesian\Coordinate Geometry we can perhaps do a similar trick. We can use line segments, and their lengths to define slope, (slope = rise over run) and then define a line as follows:

**Algebraic Definition Candidate**

A *line*, , is a set of points such that for all pairs of *distinct* points , the slope is a constant.

This means that if you take two pairs of distinct points in a line , and then calculate the slopes between them, you get the same answer, and therefore it makes sense to talk about *the *slope of a line, .

This definition, however, has exactly the same problem as the previous. The definition we use isn’t too important but I do want to use a definition that considers the line a *set of points.*

## The Equation of a Line

We can use such a definition to derive the equation of a line ‘formula’ for a line of slope containing a point .

Suppose first of all that we have an axis and a point in the line. What does it take for a second point to be in the line?

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