# Vector Equation of a Line You are probably very familiar with using y=mx + b, the slope-intercept form, as the equation of a line. While this equation works well in two-dimensional space, it is insufficient to completely define the equation of a line in higher order spaces.

Let's work in three dimensions. How much information is needed in order to specify a straight line? The answer is that we need to know two things: a point through which the line passes, and the line's direction. Both of those things can be described using vectors. So, another way to identify points on a line can be found using the vector addition method. We can add the position vector of a point on the line which we already know, for example A(a_1, a_2) and add to that a vector, vec u=(u_1, u_2). This addition gives us the Vector Equation of a Line: P = A + k vec u, k in RR or (x,y)=(a_1,a_2)+k(u_1,u_2), k in RR, as shown in the diagram below.

Created with GeoGebra by Vitor Nunes

### Interactivity

• Start by putting Point C in a place of your choice;
• Then move the vector vec (AB) until it has the desired direction;
• Click the "Connect Dot" button;
• Finally try moving the selector k.

Check that the dot cloud formed by the D point defines a line.

## straight Line in the Plane To determine a straight line in the plane, it is necessary to have two points or a point and a vector. A vector director of a straight line is any vector that has the same direction as the given straight line. With two points we can easily obtain the vector that exists between them.