Coplanarity Two Lines
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
FAQs on Coplanarity of Two Lines in 3D Geometry
1. What does it mean for two lines to be coplanar in 3D geometry?
In three-dimensional space, two lines are called coplanar if they both lie on the same single plane. Think of a flat sheet of paper; if you can draw both lines on that sheet, they are coplanar. This can happen if the lines are either parallel or they intersect at a point.
2. What is the main condition to check if two lines in 3D are coplanar?
The primary condition involves using the scalar triple product. If the first line passes through a point A and has a direction vector d₁, and the second line passes through a point B with a direction vector d₂, the lines are coplanar if the vector AB and the two direction vectors are themselves coplanar. This is true if their scalar triple product is zero.
3. If two lines are coplanar, does it mean they must intersect?
No, not necessarily. While intersecting lines are always coplanar, parallel lines are also coplanar. Both parallel lines can be placed on the same plane, but they will never meet. So, being coplanar simply means they exist on the same flat surface, which includes both intersecting and parallel cases.
4. What is the difference between coplanar lines and skew lines?
The key difference is whether the lines exist on the same plane.
- Coplanar lines lie on the same plane. They are either parallel or they intersect.
- Skew lines are non-coplanar. This means they are not parallel, and they also do not intersect. They exist in different planes and pass by each other without ever touching.
5. How can you test for coplanarity if the lines are given in Cartesian form?
If the lines are given in Cartesian form as (x - x₁)/a₁ = (y - y₁)/b₁ = (z - z₁)/c₁ and (x - x₂)/a₂ = (y - y₂)/b₂ = (z - z₂)/c₂, you check if the determinant of a specific 3x3 matrix is zero. The lines are coplanar if:
| (x₂ - x₁) (y₂ - y₁) (z₂ - z₁) |
| a₁ b₁ c₁ |
| a₂ b₂ c₂ | = 0
This calculation is the Cartesian equivalent of the scalar triple product being zero.
6. Why does the scalar triple product being zero prove that two lines are coplanar?
The scalar triple product geometrically represents the volume of a parallelepiped formed by three vectors. In this case, the three vectors are the two direction vectors of the lines and the vector connecting a point on each line. If these three vectors lie flat on a single plane (i.e., are coplanar), they cannot form a 3D shape with any volume. Therefore, a volume of zero proves they all lie on the same plane.
7. Are skew lines and perpendicular lines the same thing?
No, they are different concepts. Perpendicular lines intersect at a 90-degree angle and are therefore coplanar. Skew lines, on the other hand, never intersect and are not parallel, so they can never be on the same plane. However, it is possible for two skew lines to have their direction vectors be perpendicular.
