Vector Equation of a Line

régua e esquadro

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


Follow these steps:

  • 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

Bloco de notas

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.