5.2 The line

The simplest geometric structure (apart from the point) is the line. To describe a line $p$ completely we need to know a point $A$ which is contained by the line and a direction of the line, i.e. a non-zero vector $\vec a$. The line $p$ then consists of all points with coordinates

The number $t$ is called a parameter as it parametrises the points on the line. Note that if we bound the set of possible values of $t$ we get only parts of the line. For instance, for $t \in \langle 0,+\infty)$ we get a ray with the initial point $A$ and direction $\vec a$, whereas for $t\in\langle 0,1 \rangle$ we get a line segment with $A$ and $A + \vec a$ as its end points. This way of describing a line, i.e. by an equation (5.2), is often called the parametric equation of a line.

An alternative way of describing a line is this. A line consists of all points with coordinates $(x,y)$ which satisfy the linear equation of a line

Constants $a,b,c$ are parameters of the line. In equation (5.3), the symbols $x$ and $y$ represent unknowns. A point $(\alpha,\beta)$ is contained in the given line if and only if after substituting $\alpha$ for $x$ and $\beta$ for $y$ into (5.3) we get a valid equality ($0=0$). Let's analyze in more detail the line $p$ described by equation

The point $(1,2)$ does not lie on $p$ because after substituting to (5.4) we get $-2 = 0$ which is not true. On the contrary, $(-1,0)$ and $(0,1/2)$ after substituting give $0=0$ and so they do lie on the line. Two points are sufficient to plot a line.

We assume that the reader knows how to convert a parametric equation of a line to its linear equation and vice versa.