# Is there a point slope form for a three dimensional line?

Dec 19, 2017

Not really, but...

#### Explanation:

Something vaguely similar that you can use for a line in any number of dimensions is a point-vector form, which you could write like this:

$\underline{x} = \underline{{x}_{0}} + t \vec{v}$

In three dimensions:

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

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

where $\left({x}_{0} , {y}_{0} , {z}_{0}\right)$ is a point through which the line passes, $\left(u , v , w\right)$ is a vector describing the direction of the line and $t$ is a parameter ranging over $\mathbb{R}$.

If $\left(u , v , w\right)$ (or in more generality $\vec{v}$) is of unit length, then the parameter $t$ acquires extra meaning in being the distance along the line from the fixed point.