# What is meant by the parametric equations of a line in three-dimensional space?

Aug 20, 2014

We want to find the parametric equations of the line L passing through the point P and parallel to a vector A.

Let us take a 3-dimensional point in ${R}^{3}$, call it $P = \left({x}_{0} , {y}_{0} , {z}_{0}\right)$.

A line L is drawn such that it passes through P and is parallel to the vector $A = \left(u , v , w\right)$.

3 parametric equations can be written which express the components:

$x = {x}_{0} + t u$
$y = {y}_{0} + t v$ $\left(- \infty < t < \infty\right)$
$z = {z}_{0} + t w$

As an example, with a point $P = \left(2 , 4 , 6\right)$ and vector $A = \left(1 , 3 , 5\right)$, we have the following parametric equations:

$x = 2 + t u$
$y = 4 + 3 v$ $\left(- \infty < t < \infty\right)$
$z = 6 + 5 w$