The vector equation of a line is a different way to represent a line, and is traditionally used only in R^{3} and higher, because in R^{2} 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:

[x,y,z]=[a_{x},a_{y},a_{z}]+t[`m`_{1},`m`_{2},`m`_{3}]

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.

- construct vector PQ from points P and Q:
[Q_{x}-P_{x}, Q_{y}-P_{y}, Q_{z}-P_{z}]

- 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]

- 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