Condition for Coplanarity of Two Lines Using Vector Triple Product
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 Three Dimensional Geometry
1. What does it mean for two lines to be coplanar?
Two lines are coplanar if they lie in the same plane. In three-dimensional geometry, this means both lines can be contained within a single flat surface. Coplanar lines may:
- Intersect at a point
- Be parallel to each other
- Coincide (overlap completely)
2. How do you check if two lines are coplanar in 3D geometry?
Two lines in 3D are coplanar if the scalar triple product of their direction vectors and the vector joining a point on each line is zero. Steps to check coplanarity:
- Let direction vectors be \(\vec{a}\) and \(\vec{b}\).
- Let \(\vec{c}\) be the vector joining any point on line 1 to any point on line 2.
- Compute the scalar triple product: \(\vec{a} \cdot (\vec{b} \times \vec{c})\).
- If the result is 0, the lines are coplanar.
3. What is the condition for two lines to be coplanar?
The condition for two lines to be coplanar is that the scalar triple product equals zero. Mathematically, if lines have direction vectors \(\vec{a}\) and \(\vec{b}\), and \(\vec{c}\) joins a point on one line to a point on the other, then:
- \(\vec{a} \cdot (\vec{b} \times \vec{c}) = 0\)
4. Are parallel lines always coplanar?
Yes, parallel lines are always coplanar. Two parallel lines have the same or proportional direction vectors and can always be placed within a single plane. In three-dimensional space, parallel lines cannot be skew because a plane can always be drawn containing both lines.
5. Are intersecting lines coplanar?
Yes, intersecting lines are always coplanar. When two lines meet at a point, a unique plane can be formed that contains both lines. Therefore, any two lines that intersect must lie in the same plane.
6. What is the difference between coplanar and skew lines?
The key difference is that coplanar lines lie in the same plane, while skew lines lie in different planes. Comparison:
- Coplanar lines: May intersect or be parallel.
- Skew lines: Do not intersect and are not parallel.
- Skew lines exist only in three-dimensional geometry.
7. Can you give an example of coplanar lines?
An example of coplanar lines is two intersecting lines in the XY-plane. For example:
- Line 1: y = 2x
- Line 2: y = -x + 3
8. What is the vector form test for coplanarity of two lines?
The vector test for coplanarity uses the scalar triple product. If lines are given in vector form:
- Line 1: \(\vec{r} = \vec{a_1} + t\vec{b_1}\)
- Line 2: \(\vec{r} = \vec{a_2} + s\vec{b_2}\)
- \((\vec{a_2} - \vec{a_1}) \cdot (\vec{b_1} \times \vec{b_2})\)
9. Why are skew lines not coplanar?
Skew lines are not coplanar because no single plane can contain both lines simultaneously. In 3D space, skew lines:
- Do not intersect
- Are not parallel
- Have a non-zero scalar triple product
10. How do you prove two lines are coplanar using determinants?
Two lines are coplanar if the determinant formed by their direction vectors and connecting vector equals zero. If direction vectors are \(\vec{b_1} = (b_1,b_2,b_3)\), \(\vec{b_2} = (c_1,c_2,c_3)\), and \(\vec{a_2}-\vec{a_1} = (d_1,d_2,d_3)\), then compute:
- | b_1 b_2 b_3 |
- | c_1 c_2 c_3 |
- | d_1 d_2 d_3 |
