Coplanarity Two Lines

Coplanarity of Two Lines In 3D Geometry

Coplanar lines in 3-dimensional geometry are a common mathematical theory. To recall, a plane is 2-D in nature stretching into infinity in the 3-D space, while we have employed vector equations to depict straight lines. 

In this chapter, we will further look into what condition is mandatory to be fulfilled for two lines to be coplanar. We will learn to prove how two lines are coplanar using the condition in Cartesian form and vector form using important concepts and solved examples for your better understanding.

How do we Identify Coplanar Lines?

Why do we want for example lines m →, n and MN→MN → to be coplanar? Let’s take into account the following two cases.

(1) If m ∥n m→∥n→, then the lines are parallel and thus coplanar. Remember that, in such a case, the 3 vectors are also coplanar irrespective of the 3rd vector.

(2) Otherwise, we would require differentiating between bisecting lines (coplanar) and skew lines (not coplanar). If the lines are bisecting, then all their points will lie in the same plane as m m→ and n n→, thus MN→MN→ should lie in that same plane.

What is the Condition of Vectors Coplanarity?

• For 3-vectors: The 3 vectors are said to be coplanar if their scalar triple product equals 0. Also, if three vectors are linearly dependent, then they are coplanar.

• For n-vectors: Vectors are said to be coplanar if no more than two amongst those vectors are linearly independent.

Coplanarity of Lines Using Condition in Vector Form

Let’s take into account the equations of two straight lines as below:

• r1 = p1 + λq1

• r2 = p2 + λq2

Wondering what the above equations suggest? It implies that the 1st line crosses through a point, L, whose position vector is provided by l1 and is parallel to m1. In the same manner, the 2nd line passes through another point whose position vector is provided by l2 and is parallel to m2.

The condition for coplanarity under the vector form is that the line connecting the 2 points should be perpendicular to the product of the two vectors namely, p1 and p2. To represent this, we know that the line connecting the two said points can be expressed in the vector form as (l2 – l1). So, we have:

(l2 – l1). (P1 x p2) = 0

Coplanarity of Lines Using Condition in Vector Form

Coplanarity in Cartesian is a derivative of the vector form. Let’s take into account the two points L (a1, b1, c1) & P (a2, b2, c2) in the Cartesian plane. Let there be 2 vectors p1 and p2. Their direction ratios are provided as x1, y1, z1 and x2, y2, z2 respectively.

The vector equation of the line connecting L and P can be provided by:

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

p1 = x1i + y1j + z1k

p2 = x2i + y2j + z2k

We must now apply the above condition under the vector form in order to derive our condition in Cartesian form. By the condition stated above, the two lines are coplanar if LM. (p1 a p2) = 0. Hence, in the Cartesian form, the matrix representing this equation is provided as 0.

Solved Examples

Question 1: Prove that the Lines [a + 3]/3 = [b – 1]/1 = [c – 5]/5 and [a + 1]/ -1 = [b – 2]/2 = [c – 5]/5 are Coplanar?

Answer: On comparing the equations, we get:

[a1, b1, c1] = {-3, 1, 5} and [a2, b2, c2] = {-1, 2, 5}.

Now, using the condition of Cartesian form, we shall solve the matrix:

= 2 [5 – 10] – 1 [-15 + 5] + 0 [-6 + 1]

 = -10 + 10 = 0

Because the solution of the matrix provides a zero, we can say that the lines given are coplanar