Coplanar Vectors definition formula conditions and solved examples
The concept of coplanar vectors plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding how vectors sit within a plane is essential for solving questions in coordinate geometry, vector algebra, and even in practical scenarios in physics and engineering.
What Is Coplanar Vector?
A coplanar vector is defined as a vector that lies on the same plane as at least two other vectors. In simple words, a set of vectors are called coplanar if they all rest on a single flat surface in three-dimensional space. You’ll find this concept applied in areas such as geometry, vector algebra, and 3D physics problems.
Key Formula for Coplanar Vectors
Here’s the standard formula: For three vectors a, b, and c to be coplanar, their scalar triple product must be zero.
Coplanar Condition Formula:
This means that if you calculate the scalar triple product (dot followed by cross product), and the answer is zero, the three vectors are coplanar.
Step-by-Step Illustration
- Suppose you have vectors: a = i + j + 2k, b = 2i − j + k, c = 3i − 2j − k.
Arrange as a matrix with coefficients:| 1 1 2 |
| 2 -1 1 |
| 3 -2 -1 | - Calculate the scalar triple product (determinant):
1 [(-1 × -1) – (1 × -2)] − 1 [(2 × -1) – (1 × 3)] + 2 [(2 × -2) – (-1 × 3)]If the result was zero, the vectors would be coplanar. Here, they are not coplanar.
= 1 [1 + 2] − 1 [−2 − 3] + 2 [−4 + 3]
= 1 × 3 − 1 × (−5) + 2 × (−1)
= 3 + 5 – 2 = 6
Coplanar vs Collinear Vectors
| Coplanar Vectors | Collinear Vectors |
|---|---|
| Lie on the same plane in space. | Lie on the same line (are parallel or on top of each other). |
| At least two are linearly independent. | All are linearly dependent (multiples of one vector). |
| Example: Three direction vectors of a triangle’s sides. | Example: Force vectors along the same rope. |
Properties & Applications
- All two vectors in a plane are always coplanar.
- Three vectors are coplanar when their scalar triple product is zero.
- If more than three vectors are to be coplanar, any three taken from them should satisfy the coplanar condition.
- Coplanar vectors help in geometry (finding area, checking if points lie on the same plane) and in physics (solving force problems).
- Used in 3D geometry and computer graphics for checking if objects are flat or have volume.
Try These Yourself
- Given vectors p = 2i + j − k, q = i − 2j + k, r = i + j + ak. Find the value(s) of a for which they are coplanar.
- Check if the vectors i, j, and k are coplanar.
- Which of these sets are coplanar: (i) {2i, i + j, 3j} (ii) {i + j, i − k, j + k}?
- How are coplanar and non-coplanar vectors different in physics?
Frequent Errors and Misunderstandings
- Confusing coplanar with collinear vectors—they are not the same!
- Forgetting that only three vectors can be checked with the scalar triple product.
- Assuming all 3D vectors are coplanar by default—actually, they must satisfy the condition.
- Leaving out zero in the scalar triple product as the test of coplanarity.
Relation to Other Concepts
The idea of coplanar vectors connects closely with vector algebra and collinear vectors. Mastery of coplanar vectors makes it much easier to solve problems involving the scalar triple product and 3D geometry involving planes and lines.
Cross-Disciplinary Usage
Coplanar vectors are not only useful in Maths but also play important roles in Physics, Engineering (like statics and mechanics), and Computer Graphics. For students preparing for exams like JEE, NEET, or other competitive tests, knowing how to test for coplanarity can quickly solve multiple-choice and assertion-reasoning type questions. Vedantu Live Classes often use real-life analogies (like the blades of a paper fan all lying in a plane) to help students visualize this concept.
Classroom Tip
A quick way to remember the coplanarity condition is “Triple product zero means the vectors are in the same plane.” Draw three pencils or rulers so their tips touch the table—no matter the direction, if they rest flat, they are coplanar!
We explored coplanar vectors—from definition, formula, calculation steps, and error traps, to how they link to other vector concepts. You can dive deeper into related topics such as properties of vectors or study more advanced geometric applications by exploring Vedantu’s Coplanarity of Vectors page.
FAQs on Coplanar Vectors in Three Dimensional Geometry
1. What are coplanar vectors?
Coplanar vectors are vectors that lie in the same plane or can be contained within a single plane in three-dimensional space. In simple terms, if you can draw all the vectors on one flat surface without lifting your pencil, they are coplanar vectors. In 2D geometry, all vectors are automatically coplanar. In 3D, vectors are coplanar if one of them can be expressed as a linear combination of the other two.
2. How do you check if three vectors are coplanar?
Three vectors are coplanar if their scalar triple product is equal to zero. For vectors a, b, and c, check:
a · (b × c) = 0
Steps to verify:
- Find the cross product b × c.
- Take the dot product of a with the result.
- If the value is 0, the vectors are coplanar.
3. What is the condition for coplanar vectors?
The condition for coplanar vectors is that their scalar triple product equals zero. Mathematically, for vectors a, b, c:
a · (b × c) = 0
This means the volume of the parallelepiped formed by the three vectors is zero, so they lie in the same plane.
4. What is the formula of the scalar triple product?
The scalar triple product of three vectors is given by a · (b × c). If a = (a₁, a₂, a₃), b = (b₁, b₂, b₃), and c = (c₁, c₂, c₃), then:
a · (b × c) = | a₁ a₂ a₃
b₁ b₂ b₃
c₁ c₂ c₃ |
This determinant form is commonly used to test coplanarity and to calculate the volume of a parallelepiped.
5. Are all 2D vectors coplanar?
Yes, all vectors in two-dimensional space are always coplanar because they lie in the same plane. Since 2D vectors have only x and y components, they cannot extend outside the plane. Coplanarity becomes meaningful mainly in three-dimensional vector geometry.
6. Can you give an example of coplanar vectors?
An example of coplanar vectors is a = (1, 0, 0), b = (0, 1, 0), and c = (1, 1, 0).
- Compute b × c = (0, 0, -1).
- Now calculate a · (b × c) = (1,0,0) · (0,0,-1) = 0.
7. What is the geometric meaning of coplanar vectors?
Geometrically, coplanar vectors are vectors that lie on the same flat surface (plane) in space. If three vectors are not coplanar, they form a three-dimensional shape with volume. If they are coplanar, the volume of the parallelepiped formed by them is zero, meaning they do not extend into three dimensions.
8. What is the difference between coplanar and collinear vectors?
The main difference is that collinear vectors lie on the same line, while coplanar vectors lie in the same plane.
- Collinear vectors: One vector is a scalar multiple of the other.
- Coplanar vectors: One vector can be written as a linear combination of two others.
9. How are coplanar vectors related to linear dependence?
Three vectors are coplanar if they are linearly dependent. This means one vector can be expressed as:
c = x a + y b
for some scalars x and y. Linear dependence implies the scalar triple product is zero, confirming coplanarity in 3D vector space.
10. Why is the scalar triple product zero for coplanar vectors?
The scalar triple product is zero for coplanar vectors because it represents the volume of the parallelepiped formed by the three vectors. When vectors lie in the same plane, they cannot enclose any three-dimensional space, so the volume is 0. Hence, a · (b × c) = 0 confirms coplanarity.
