# What is a line (coordinate geometry)

1. Jul 23, 2014

### 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).

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