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What is a line (coordinate geometry)

  1. Jul 23, 2014 #1
    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
    [itex]a_1(x_1-p_1)\,=\, a_2(x_2-p_2)\,=\,\cdots\,=\,a_n(x_n-p_n)[/itex]
    where [itex]\,a_1,p_1,a_2,p_2,\cdots a_n,p_n\,[/itex] are constants.

    Equations

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

    [tex]\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}[/tex]

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

    [tex]y\,=\,kx\,+\,n[/tex] (1)

    [tex]Ax\,+\,By\,+\,C\,=\,0[/tex] (2)

    [tex]A(x-p_1)\,+\,B(y-p_2)\,=\,0[/tex]

    [itex]k[/itex] , or [itex]\frac{-A}{B}[/itex] , 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 [itex]\alpha[/itex] in Picture #1)

    If [itex]n\,=\,0[/itex] , or [itex]C\,=\,0[/itex] , then the Line goes through the origin of coordinates [itex](0,0)[/itex] (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 [itex]n \neq 0[/itex], 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, [itex]B \neq 0[/itex], the equation (2) is Ax+By=0, or [itex]y=-\frac{A}{B}x[/itex] (line which is passing across the point (0,0) - Picture #2);

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

    g)If A=0 and [itex]B \neq 0[/itex], then By+C=0, or [itex]y=-\frac{C}{B}[/itex], or [itex]-\frac{C}{B}=n[/itex], 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!
     
  2. jcsd
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