The vector equation of a line is a different way to represent a line, and is traditionally used only in R3 and higher, because in R2 there are easier ways of writing it.

For those of you who just can't wait to get your hands on this piece of math, the equation looks something like this:


What this means is that any point p(x,y,z) on a line can be represented by a position, vector a (in Cartesian form), plus a parameter t times a direction vector m

The easiest way to derive this equation is from two points on the line, P and Q.

  1. construct vector PQ from points P and Q:
    [Qx-Px, Qy-Py, Qz-Pz]
  2. Take point Q or P (whichever is simpler) and use it as your position vector: point Q(x,y,z) becomes vector a[x,y,z]

  3. Combine the two to form your completed equation:
    [x,y,z]=a + t PQ

It may be interesting to note that this equation requires you to break the rules regarding vectors and points in a certain sense. The result of the equation is a vector, but you use it as a point (because the components of the vector correspond to points on the line, and in addition, you use a point as a vector when constructing the equation (step 2).

This form of a line can be extended into as many dimensions as needed (as it can also be used in two-space). The parametric equations of a line and symmetric equations of lines are also derived from this form.

You may also be interested in the vector equation of a plane