Straight line equations

The direction vector for a line is any vector with the same direction as the straight line.

To determine a straight line and its equations, we need to know one point and a direction vector or two points (to obtain a direction vector).

If we have a point, A(a₁,a₂), and a direction vector, u(u₁,u₂), of a straight line r, then any point X Є r, has a position vector X(x,y):

In coordinates:

If we work out λ:

By doing the cross-product and simplifying:

The slope or gradient of a straight line, m, is the tangent of the angle that the line and the abscissa axis form : m = tan α

You can see that:

Then, a straight line can be determined too, if we know a point and the slope:

NOTE:

Example: find out all the equations of a straight line that passes through the point A(1,1) in the direction of the vector u(-3,2).

 

Exercises:

1.- Find out all the equations of a straight line that passes through the point (3,-1) and has the direction of the vector (-1,1).

2.- Find the other equations of the straight line r: y = 3x + 2

 

 

Solutions: 

1.-

 

2.-