# 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*(a_{1},a_{2}), and a direction vector, *u*(u_{1},u_{2}), 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.-

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