5. Problems with lines in parametric form II
Block :Geometry
5.3. RELATIVE POSITIONS OF TWO LINES
Given the lines

r1: 

r2: 

 to find their relative position, we solve the following simultaneous equations in two unknowns, t and s:
We equate the x and the y of the two lines using different parameters, t and s, one for each.
If the equations have only one solution (t0,s0), the lines cut at a point, its coordinates can be found by substituting ,t for t0 in r1, and also, t for s0 in r2,   .
If the solutions do not have a solution, the lines are parallel.
If the equations have infinite solutions, they are of the same line.

The lines which appear at the start of the figure are:   and 
In it we can change the values of a, b, c and d, which correspond to the line r1, that is, the point on r1 is (a,c)=(5,0) and its direction vector is (b,d)=(-1,3)

1.- Equating the x and the y of the two equations, giving the label s to the parameter of r2. Solve the resulting simultaneous equations. There should be only one solution for t and s. 

2.- Substituting t into r1 or s into r2 to find the point P of intersection of the two lines. You have the solution in the figure. Verify it. 

You can type in the new values and press enter.

3.- Using the buttons at the bottom of the figure change the value of b, put b=2, and for d, put d= -3. What have we changed in the line r1?

4.- Comment on r1 and r2 now. Solve the new simultaneous equations as proof. 

5.- Now put a=1, b=-6, c=3 and d=9 What has occurred? Now solve the simultaneous equations. 

6.- Change the values of a, b, c and d, and you are going to see the effect in the figure. 

       
           
  Ángela Núñez Castaín
 
Ministry of Education , Social Afairs and Sport. Year 2001
 
 

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