Parametric form of a line

Question

I've often seen that a line can be expressed in a parametric form

$x = x_0 + t.( x_1 − x_0 )$

$y = y_0 + t.( y_1 − y0 )$

But I can't see how this makes sense. Would anyone be able to explain me the meaning of each of these equations?

Answer 1

The parametric equations of a line express the fact that given any three points $P$, $Q$ and $R$ on it, the vectors $\vec{PQ}$ and $\vec{PR}$ are parallel, i.e. $$ \vec{PR}=t\cdot\vec{PQ} $$ for some $t\in\Bbb R$. If you rewrite the displayed equation in terms of the coordinates of $P$ and $Q$ (to be thought "fixed") and $R$ ("variable" or "moving"), you get the parametric equations.

Note that this is valid for a line in ${\Bbb R}^n$ for any $n$.

Answer 2

I'll go over the equation of a straight line on $\mathbb R^2$ and not $\mathbb R^3$, because in 3-dimensions things get a bit more complicated and require you to have understood geometric properties of equations.

First of all, after we assume the Cartesian System $(x,y)$, let's start with the simplest equation possible :

$$y - 5 = 0\Leftrightarrow y=5$$

To understand its geometric properties, let's rewrite :

$$y + 0x = 5$$

Now, we can see that $\forall x \in \mathbb R$, $y=5$.

So, that tells us that regardless the number of $x$, the variable $y$ is fixed on a constant number, which means no variations in the form of your geometric object/line.

The constant representation of such an occasion is presented in the $x-y$ coordinate system as a straight line, parallel to the $x'x$ axis.

Things get a bit more complicated though, when we have a $2$ variable equation with non-negative coefficients :

$$x + y + 5 = 0 \Leftrightarrow y = -x-5 $$

Let's observe that for any given $x$, $y$ has a unique value. Thus, every pair of $(x,y$) is unique and can be dotted over the $x-y$ plane. An infinite "drawing" of these dots, that represent every pair $(x,y)$ that satisfies the equation, will result in a continuous straight line.

Studying the properties of such an equation, is though essential. Let's now assume the general parametric equation :

$$y-y_0 = λ(x-x_0) \Leftrightarrow y =λx - λx_0+y_0$$

Observe that $x,y$ are variables but that $λ,x_0,y_0$ are constant values. Then, our equation could be re-written as :

$$y = λx +C$$

which resembles the form we had in the previous example.

The number $λ\in \mathbb R$ determines the orientation of the line, since as you probably know, from any given point, infinite lines can be crossing it.

The constant value $C\in \mathbb R$ is just a numerical value that determines the position of the line with respect to $(0,0)$. If $C=0$ then the line crosses by $(0,0)$, otherwise not.

Since we are done with explaining the equation of the line and making you understand why it is like that, let's form a more straigh-up mathematical explanation.

Let's assume any three points on the $x-y$ plane : $A,B,C$. The parametric equation of the line, expresses the fact that the vectors created with the same starting point $A$ : $\vec{AB}$ and $\vec{AC}$, are parallel, which in vector calculus is expressed as :

$$\vec{AB} = λ \cdot \vec{AC}$$

for some $λ\in\mathbb R$.

Let's say that : $A=(x_a,y_a),B=(x_b,y_b),C=(x_c,y_c)$. Then the equation above is written as :

$$(x_b - x_a, y_b - y_a) = λ(x_c - x_a, y_c - y_a)$$

$$\Leftrightarrow$$

$$\{x_b - x_a = λ(x_c - x_a),y_b - y_a = λ(y_c - y_a)\}$$

which eventually are, the equations you're asking for.