Condition for Coplanarity of Two Lines in 3D Geometry with Formula and Examples
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 of Two Lines in Three Dimensional Geometry
1. What does coplanarity of two lines mean?
Two lines are coplanar if they lie in the same plane in three-dimensional space. This means:
- Both lines can be drawn on a single flat surface (plane).
- They may intersect, be parallel, or even coincide.
- If two lines are not in the same plane and do not intersect, they are called skew lines.
Coplanarity is an important concept in coordinate geometry and 3D vector algebra.
2. How do you check if two lines are coplanar in 3D?
Two lines in 3D are coplanar if the scalar triple product of their direction vectors and the vector joining any point on one line to a point on the other line is 0. Steps:
- Find direction vectors d₁ and d₂ of the two lines.
- Take a point A on line 1 and B on line 2.
- Form vector AB.
- Compute scalar triple product: d₁ · (d₂ × AB).
If the result equals 0, the lines are coplanar; otherwise, they are skew.
3. What is the condition for two lines to be coplanar?
The condition for coplanarity of two lines is that the scalar triple product equals zero. Mathematically, if d₁ and d₂ are direction vectors and AB is the vector joining points on the lines, then:
- d₁ · (d₂ × AB) = 0
This condition ensures that the three vectors lie in the same plane, confirming the lines are coplanar.
4. Are parallel lines always coplanar?
Yes, two parallel lines are always coplanar because there exists exactly one plane that contains both lines. In geometry:
- Parallel lines have proportional direction vectors.
- They never intersect.
- They lie on the same flat surface (plane).
Thus, parallel lines in 3D space are a special case of coplanar lines.
5. What is the difference between coplanar and skew lines?
Coplanar lines lie in the same plane, while skew lines lie in different planes and do not intersect. Key differences:
- Coplanar lines: May intersect or be parallel.
- Skew lines: Non-parallel and non-intersecting.
- Coplanarity condition: scalar triple product = 0.
This distinction is important in 3D coordinate geometry.
6. Can two intersecting lines be coplanar?
Yes, two intersecting lines are always coplanar because exactly one plane passes through two intersecting lines. In 3D geometry:
- If two lines intersect at a point, they share that point.
- A unique plane can be formed using the two lines.
Therefore, intersecting lines automatically satisfy the coplanarity condition.
7. What is a simple example of checking coplanarity of two lines?
Two lines are coplanar if the scalar triple product is 0. Example:
- Line 1 direction vector: d₁ = (1, 0, 0)
- Line 2 direction vector: d₂ = (0, 1, 0)
- Let AB = (0, 0, 1)
Compute d₂ × AB = (1, 0, 0).
Then d₁ · (d₂ × AB) = (1,0,0) · (1,0,0) = 1.
Since the result is not 0, the lines are not coplanar (they are skew).
8. Are all lines in a plane coplanar?
Yes, all lines drawn within a single plane are coplanar by definition. In plane geometry:
- Any number of lines can lie in the same plane.
- They may intersect, be parallel, or coincide.
The concept of non-coplanar (skew) lines only arises in three-dimensional space.
9. How do you prove two lines are not coplanar?
Two lines are not coplanar if the scalar triple product is non-zero. To prove:
- Find direction vectors d₁ and d₂.
- Find vector AB joining points on the lines.
- Compute d₁ · (d₂ × AB).
If the result ≠ 0, the lines are skew lines, meaning they do not lie in the same plane.
10. Why is coplanarity important in coordinate geometry?
Coplanarity is important because it helps determine whether two lines intersect, are parallel, or are skew in 3D coordinate geometry. It is used to:
- Identify relationships between lines in space.
- Solve problems involving shortest distance between skew lines.
- Analyze geometric structures in engineering and physics.
Understanding coplanarity ensures accurate spatial reasoning and vector calculations.
