Coplanarity of Two Lines in 3D Geometry

The lines that lie on the same plane are what we call the Coplanar lines. Coplanar lines are a common concept with regards to 3-dimensional geometry. Call to mind, a plane is a 2-dimensional figure stretching out into infinity in the 3-dimensional space, while we have taken into application vector equations in order to represent lines or straight lines. That said, given two lines L1 and L2, each crossing through a point whose position vector are provided as (A, B, C) and parallel to line whose direction ratios are given as (X, Y, Z), the task is to look over if line L1 and L2 are coplanar or not.

Conditions to Prove Coplanarity of Two Lines

Let’s now have a look at what condition is mandatory to be fulfilled for two lines to be coplanar. From mathematical concepts, we may describe coplanarity as the condition where a given number of lines are located on the same plane, and are said to be coplanar. Under 3-dimensional geometry we can use the condition in cartesian form and vector form in order to prove that two lines are coplanar.

Condition for Coplanarity Using Cartesian Form

The condition for coplanarity in the Cartesian form emerges from the vector form. Let us  consider two points L (a1, b1, c1) & Q (a2, b2, c2) in the Cartesian plane. Assume that there are two vectors q1 and q2. Their direction ratios are provided by x1, y1, z1 and x2, y2, z2 respectively.

The vector form of equation of the line connecting L and Q can be given as under:

LQ = (a2 – a1)i + (b2 – b1)j + (c2 – c1)k

Q1 = x1i + y1j + z1k

Q2 = x2i + y2j + z2k

We now use the above condition in a vector in order to induce our condition in Cartesian form. This can be put to application for the calculation purpose. From the above condition, the two lines are said to be coplanar if LQ. (Q1 x Q2) = 0. Hence, in Cartesian form, the matrix illustrating this equation is given as 0.

Condition for Coplanarity Using Vector Form

For the derivation of the condition for coplanarity in vector form, we shall take into consideration the equations of two straight lines to be as below:

r1 = l1 + λq­1

r2 = l2 + λq2

You must be thinking what these above equations mean? Well! It means that the 1st line crosses through a point, say L, whose position vector is provided by L1 and is running parallel to q1. In the same manner, the 2nd line passes through another point whose position vector is provided by L2 and runs parallel to q2.

The condition for coplanarity in vector form is that the line connecting the two points should be perpendicular to the product of the two vectors, q1 and q2. To depict this, we know that the line connecting the two said points can be mathematically expressed in vector form as (L2 – L1). Thus, we have:

(L2 – L1). (Q1 x Q2) = 0

Solved Examples For You To Prove Coplanarity Of Two Lines

Example 1:

Input:

L1: (a1, b1, c1) = [-2, 2, 4] and (x1, y1, z1) = [-2, 2, 4]

L2: (a1, b1, c1) = [-2, 2, 4] and (x1, y1, z1) = [-2, 2, 4]

Output: Lines are said to be coplanar since lie in the same plane

Example 2:

Input:

L1: (a1, b1, c1) = [1, 2, 4] and (x1, y1, z1) = [2, 5, 4]

L2: (a1, b1, c1) = [-1, 3, 4] and (x1, y1, z1) = [6, 1, 5]

Output: The two lines do not lie in a same plane, thus they are NOT coplanar