What is a line (coordinate geometry)

[Greg Bernhardt]

Definition/Summary

In Euclidean coordinate geometry, a Line usually means the whole (infinitely long) Line.

(In ordinary Euclidean geometry, a Line usually means a line segment between two points.)

The equation of a Line in n dimensions is a combination of n-1 linear equations of the form
$a_1(x_1-p_1)\,=\, a_2(x_2-p_2)\,=\,\cdots\,=\,a_n(x_n-p_n)$
where $\,a_1,p_1,a_2,p_2,\cdots a_n,p_n\,$ are constants.

Equations

Through every pair of points $(p_1,\,p_2,\,\cdots\ p_n)$ and $(q_1,\,p_2,\,\cdots\ p_n)$ in n dimensions, there is a unique Line:

$$\frac{(x_1-p_1)}{q_1-p_1}\,=\,\frac{(x_2-p_2)}{q_2-p_2}\,=\,\cdots\,=\,\frac{(x_n-p_n)}{q_n-p_n}$$

In a plane, the equation of a Line is usually written in one
of three ways:

$$y\,=\,kx\,+\,n$$ (1)

$$Ax\,+\,By\,+\,C\,=\,0$$ (2)

$$A(x-p_1)\,+\,B(y-p_2)\,=\,0$$

$k$ , or $\frac{-A}{B}$ , is the Gradient of the Line, and is the tangent of the angle measured anti-clockwise from the positive x-axis to the Line (the angle $\alpha$ in Picture #1)

If $n\,=\,0$ , or $C\,=\,0$ , then the Line goes through the origin of coordinates $(0,0)$ (see Picture #2).

In a plane, every pair of Lines (other than a parallel pair) meet at a unique point.

Extended explanation

Let's look for some special cases of the equation (1):

a)If k=1 and n=0, the equation will come to y=x, and it is symmetrical line of the first and the third quadrant;

b)If k=-1 and n=0, the equation (1) will come to y=-x, and it is symmetrical line of the second and the fourth quadrant;

c)If k=0 and $n \neq 0$, the equation (1) will be y=n, and it is line parallel to the x-axis;

d)If k=0 and n=0, the equation (1) is y=0 which is equation of the x-axis.

Let's look for some special cases of the equation (1):

e) C=0, $B \neq 0$, the equation (2) is Ax+By=0, or $y=-\frac{A}{B}x$ (line which is passing across the point (0,0) - Picture #2);

f) If B=0 and $A \neq 0$, then the equation of line (2) is Ax+C=0.
Solving for x, we get $x=-\frac{C}{A}$ or $-\frac{C}{A}=a$, x=a (this line is parallel to the Oy axis);

g)If A=0 and $B \neq 0$, then By+C=0, or $y=-\frac{C}{B}$, or $-\frac{C}{B}=n$, y=n (this line is parallel to the Ox axis - Picture #3).

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!

 

[fresh_42]

It is interesting that we still teach the Euclidean view of space: "The shortest distance between two points is a line." Wrong! The shortest distance between two points is the geodesic! It only happens that in Euclidean, that is absolutely flat spaces the lines are the geodesics. But there is nothing in our cosmos which is absolutely flat, only approximately flat.