# Coplanarity

In geometry, some points in space are coplanar if there is a geometric plane that includes them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. However, a set of four or more distinct points will, in general, not lie in a single plane.

Two lines in three-dimensional space are coplanar if there is a plane that includes them both. This occurs if the lines are parallel, or if they intersect each other.

Distance geometry provides a solution technique for the problem of determining if a set of points is coplanar, knowing only the distances between them.

## Properties

In three-dimensional space, two independent vectors with the same initial point determine a plane through that point. Their cross product is a normal vector to that plane and any vector orthogonal to this cross product through the initial point will lie in the plane.[1] This leads to the following coplanarity test. Four distinct points, x1, x2, x3 and x4 are coplanar if and only if,

$(x_3 - x_1) \cdot [(x_2 - x_1) \times (x_4 - x_3)] = 0.$

If three vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are coplanar, then

$(\mathbf{c}\cdot\mathbf{\hat a})\mathbf{\hat a} + (\mathbf{c}\cdot\mathbf{\hat b})\mathbf{\hat b} = \mathbf{c},$

where $\mathbf{\hat a}$ denotes the unit vector in the direction of $\mathbf{a}$. That is, the vector projections of $\mathbf{c}$ on $\mathbf{a}$ and $\mathbf{c}$ on $\mathbf{b}$ add to give the original $\mathbf{c}$.

## Coplanarity of points whose coordinates are given

In coordinate geometry, in n-dimensional space, a set of four or more distinct points are coplanar if and only if the matrix of the coordinates of these points is of rank 2 or less. For example, given four points, W = (w1, w2, ... , wn), X = (x1, x2, ... , xn), Y = (y1, y2, ... , yn), and Z = (z1, z2, ... , zn), if the matrix

$\begin{bmatrix} w_1 & w_2 & \dots & w_n \\ x_1 & x_2 & \dots & x_n \\ y_1 & y_2 & \dots & y_n \\ z_1 & z_2 & \dots & z_n \end{bmatrix}$

is of rank 2 or less, the four points are coplanar.