# Dictionary Definition

coplanar adj : lying in the same plane

# User Contributed Dictionary

## English

### Adjective

- (Of at least two things, usually lines) within the same plane

### See also

# Extensive Definition

In geometry, a set of points in space is coplanar
if the points all lie in the same geometric plane.
For example, three distinct points are always coplanar; but four
points in space are usually not coplanar.

Points can be shown to be coplanar by determining
that the scalar
product of a vector
that is normal to
the plane and a vector from any point on the plane to the point
being tested is 0. To put this another way, if you have a set of
points which you want to determine are coplanar, first construct a
vector for each point to one of the other points (by using the
distance
formula, for example). Secondly, construct a vector which is
perpendicular
(normal) to the plane to test (for example, by computing the
cross
product of two of the vectors from the first step). Finally,
compute the dot product
(which is the same as the scalar product) of this vector with each
of the vectors you created in the first step. If the result of each
dot product is 0, then all the points are coplanar.

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

## Properties

If three 3-dimensional vectors \mathbf, \mathbf and \mathbf are coplanar, and \mathbf\cdot\mathbf = 0, then- (\mathbf\cdot\mathbf)\cdot\mathbf + (\mathbf\cdot\mathbf)\cdot\mathbf = \mathbf,

where \mathbf denotes the unit vector
in the direction of \mathbf.

Or, the vector
resolutes of \mathbf on \mathbf and \mathbf on \mathbf add to
give the original \mathbf.

## External links

coplanar in Bulgarian: Компланарност

coplanar in German: Komplanarität

coplanar in Russian: Компланарность

coplanar in Slovenian:
Komplanarnost