Coplanar Geometry

Three or more points, lines or any other geometric shapes that lie on the common plane are knows as Coplanar.

Geometric substance lying in a same plane are said to be coplanar. In a plane three noncollinear points are some extent coplanar. Four points are coplanar, defined by them is 0, lying in the same plane. Example, any set of three points in plane are coplanar. Let us see coplanar geometry in brief.

Conditions for coplanar:

Coplanarity is corresponding to the statement that the pair of lines determined by the four points is not skew, and can be equivalently stated in vector form as

`|[x1,y1,z1,1],[x2,y2,z3,1],[x3,y3,z3,1],[x4,y4,z4,1]| = 0`

(x3 - x1).[(x2 - x1) × (x4 - x3)] = 0

The coplanar not only for four points it is also for two or three points.

An arbitrary number of n  points x1 , ..., xn can be checked for coplanarity by finding the point-plane distances of the points x4, ...,xn from the plane determined by (x1,x2,x3), and checking if they are all zero. Therefore, the points are all coplanar.

A set of n vectors v is coplanar if the nullity of the linear mapping defined by v  has dimension 1, the matrix rank of v (or equivalently, the number of its singular values) is n-1.

Parallel lines in three-dimensional space are said to be coplanar, but skew lines are not.

In this article we see the coplanar of lines on same plane. Using the line equation we find the coplanar for the three lines on a same plane.

Example problem for coplanar:

Example: prove that three lines are coplanar, equation lines are 3x + 2y = 0, 3x + 3y = 3, 2x + 2y = 2

Solution:

Given: 3x + 2y = 0 -------------(1)

3x + 3y = 3 -------------(2)

2x + 2y = 2 --------------(3)

`|[x1,y1,c],[x2,y2,c],[x3,y3,c]| = 0`

`|[3,2,0],[3,3,3],[2,2,2]| = 0`

3[(3×2) - (3×2)] - 2[(3×2) - (3×2)] +0[(3×2) - (3×2)] = 0

3(6 - 6) -2(6 - 6) -0(6 - 6) =0

0 = 0

Hence proved thus the three lines are coplanar.