# How do I determine whether a line is in a given plane in three-dimensional space?

Sep 17, 2015

For plane:

$a x + b x + c x + d = 0$

and line:

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

these are suitable conditions:

{ (ax_0+by_0+cz_0+d = 0), (au+bv+cw = 0) :}

#### Explanation:

The most general equation of a plane in three dimensional space is:

$a x + b y + c z + d = 0$

A line in three dimensional space can be represented in a variety of ways, but one representation that will work for any line is the parametric form:

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

This line will lie in the plane if and only if two distinct points of it both satisfy the equation of the plane.

Using $t = 0$ and $t = 1$, we obtain the conditions:

{ (ax_0+by_0+cz_0+d = 0), (a(x_0+u) + b(y_0 + v) + c(z_0 + w) + d = 0) :}

Subtracting the first of these from the second, we obtain the condition:

$a u + b v + c w = 0$

So a necessary and sufficient set of conditions is:

{ (ax_0+by_0+cz_0+d = 0), (au+bv+cw = 0) :}