3. Equations of the straight line I
Block :Geometry
 
3.1. EQUATIONS OF THE STRAIGHT LINE: VECTOR EQUATION 

A line r can be found using vectors in the following way: 

  • Given a point P on the line, assuming that it gives the vector OP =p, called the position vector.
  • Given a vector, d, parallel to the line called the vector of direction.
    • 1.-In the adjoining figure, change the value of the parameter t and observe the vectors of origin O, p, p+d, p+2d, p-d, ... all of these have their furthest point on the line r. 
    In general, p+td is a vector which, if situated with its origin at O, has its furthest point, X, on the line r and slides on top of it at the variable t.

    This gives us the vector equation of the straight line:

    OX = p + t.d

    O   is the origin of the coordinates 
    X    is any variable point on the line 
    p   is the position vector of a known point P on the line 
    d   is a known vector of direction, parallel to the line 
    t   is a parameter. By giving values to t, we will obtain the different points X on the line
    3.2. EQUATIONS OF THE LINE: PARAMETRIC eQUATIONS  AND THE GENERAL EQUATION

    If the vectors are substituted into the vector equation by their coordinates, this follows: 

    (x,y) = (p1,p2) + t (d1,d2)

    Expressing each coordinate separately the parametric equations are obtained:
    (x,y)   are the coordinates of any unknown point on the line 
    (p1,p2)   are the coordinates of a known point on the line 
    (d1,d2)   are the coordinates of a vector parallel to the line 
    t  is a parameter. For each value that we give to t a point (x,y) on the line is obtained.

    If in the parametric equations we eliminate the parameter (for example, clearing t in one of them and substituting its value in the other), a unique equation called the general equation is obtained. 

    Ax + By + C = 0
    1.- Copy into your workbook and with the aid of this figure understand the following process to find the equations of the line  r.

    We take: 

    • the position vector of any point on r  (p(3,6))
    • any vector, parallel to r (d(3,2))
    In this figure if you are changing the value of the parameter t, you will be seeing different points X on the line r, and the resulting coordinates are those of the parametric equations.
    The different equations of the line r are: PARAMETRIC EQUATIONS

    We remove  t, multiply the first equation by 2, the second by -3 and we sum them obtaining the equation: IMPLICIT EQUATION  2x - 3y +12=0  .
           
               
      Ángela Núñez Castaín
     
    Ministry of Education, Social Afairs and Sport. Year 2001
     
     

    Licencia de Creative Commons
    Except where otherwise noted, this work is licensed under a Creative Common License