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
FAQs on Coplanarity Two Lines
1. What does it mean for two lines to be coplanar in 3D geometry?
Two lines in three-dimensional space are called coplanar if they both lie on the very same flat surface, or plane. Think of a blackboard: any two lines you draw on it are coplanar. If one line is on the blackboard and another line passes through the air in front of it, they are non-coplanar.
2. What is the main condition used to check if two lines are coplanar?
The main condition involves vectors. Let's say the first line passes through a point A (with position vector a₁) and is parallel to a vector b₁. The second line passes through a point C (with position vector a₂) and is parallel to a vector b₂. The two lines are coplanar if the vector connecting the two points and the two direction vectors lie on the same plane, which means their scalar triple product [(a₂ - a₁) b₁ b₂] must be 0.
3. How are non-coplanar lines different from coplanar lines?
The key difference is that coplanar lines exist on the same plane, while non-coplanar lines do not. Non-coplanar lines that do not intersect are also known as skew lines. For example, the top edge of a book and its spine are intersecting, making them coplanar. However, the top-front edge and the back-bottom edge of the book are non-coplanar because they are not parallel and will never meet.
4. Why is the concept of coplanarity important specifically in three-dimensional (3D) space?
In 2D geometry, like on a flat sheet of paper, all lines are automatically on the same plane, so the concept isn't needed. In 3D space, however, lines can exist on countless different planes. Determining if two lines are coplanar is a crucial first step to understand their relationship. For instance, you can only find a point of intersection for two lines after confirming they are coplanar. If they are not coplanar, they cannot intersect.
5. Can you provide a simple real-world example of coplanar lines?
Certainly. Imagine the lines on a ruled notebook page. All the horizontal lines printed on that page are coplanar because they all lie on the flat surface of the paper. A good example of non-coplanar lines would be the flight path of an aeroplane and a straight road it flies over—they exist in different planes and do not intersect.
6. How is the condition for coplanarity applied to solve problems, like finding an unknown variable?
This is a common type of problem in Class 12 exams. If you are given the equations of two lines and are told they are coplanar, you can use this fact to find a missing value (like 'k' or 'λ'). You would set up the scalar triple product of the relevant vectors and set the result equal to zero. This creates an equation where the unknown variable is the only thing left to solve for.
7. Does the method to check for coplanarity change if the lines are given in vector form versus Cartesian form?
The underlying principle remains the same, but the steps look a bit different.
- Vector Form: This is more direct. You identify the points and direction vectors from the equations r = a₁ + λb₁ and r = a₂ + μb₂ and apply the scalar triple product test: (a₂ - a₁) . (b₁ x b₂) = 0.
- Cartesian Form: You first extract the coordinates of the points and the direction ratios from the equations. Then you set up a determinant using these values and check if it equals zero. While the calculation looks different, it is mathematically the same as the scalar triple product.
8. Are intersecting or parallel lines in 3D space always coplanar?
Yes, absolutely. This is a fundamental geometric rule to remember.
- Intersecting Lines: If two lines intersect, they must lie on the same plane that contains both of them. The point of intersection helps define this common plane.
- Parallel Lines: Similarly, two distinct parallel lines will always lie on a single, unique plane.
