# What is the symmetric equation of a line in three-dimensional space?

Nov 20, 2014

The symmetric equation of the line with the direction vector $\vec{v} = \left(a , b , c\right)$ passing through the point $\left({x}_{0} , {y}_{0} , {z}_{0}\right)$ is:

$\frac{x - {x}_{0}}{a} = \frac{y - {y}_{0}}{b} = \frac{z - {z}_{0}}{c}$,

where none of $a , b$ and $c$ are zero.

If one of $a , b$, and $c$ is zero; for example, $c = 0$, then we can write:

$\frac{x - {x}_{0}}{a} = \frac{y - {y}_{0}}{b}$ and $z = {z}_{0}$.

If two pf $a , b$, and $c$ are zero; for example, $b = c = 0$, then we can write:

$y = {y}_{0}$ and $z = {z}_{0}$

(There is no restriction on $x$, it can be any real number. )

I hope that this was helpful.