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

\begin{equation}\label{eq-rce-primky}\tag{5.2} (x,y) = A + t\cdot\vec a, \quad t\in\R. \end{equation}

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

\begin{equation}\label{eq-rceprimky}\tag{5.3} ax + by + c = 0. \end{equation}

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

\begin{equation}\label{eq-primka-ex}\tag{5.4} x - 2y + 1 = 0. \end{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.

Question 5.1

Convert the parametric equation of a line into a linear equation: $(x,y) = (1,2) + (2t,-t)$, $t\in\mathbb{R}$.

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Question 5.2

Convert the linear equation into a parametric equation for $3x-2y+1=0$.

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$(x,y) = (-1,-1) + t\cdot(2,3)$.

Question 5.3

Construct a linear equation of a line containing points $(1,-3)$ and $(2,4)$.

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$7x - y - 10 = 0$.