How to Prove Three Vectors are Coplanar?
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 Explained: Meaning, Formula & Key Examples
1. What are coplanar vectors?
Coplanar vectors are vectors that lie on the same plane in three-dimensional space. This means they are all parallel to a single plane. Any two vectors are always coplanar, as they can always be considered to lie on a single plane.
2. What is the condition for three vectors to be coplanar?
Three vectors are coplanar if their scalar triple product is zero. The scalar triple product is calculated as the determinant of the matrix formed by the vectors' components. If this determinant is 0, the vectors lie on the same plane.
3. How do you prove vectors are coplanar?
To prove coplanarity, you typically calculate the scalar triple product of the vectors. If the result is zero, then the vectors are coplanar. Alternatively, you can show that one vector can be expressed as a linear combination of the others.
4. What is the difference between coplanar and collinear vectors?
Collinear vectors lie on the same line, while coplanar vectors lie on the same plane. All collinear vectors are coplanar, but not all coplanar vectors are collinear.
5. Can two vectors be coplanar?
Yes, any two vectors are always coplanar. They can always be considered to lie within a single plane.
6. How does the scalar triple product help check coplanarity?
The scalar triple product of three vectors, represented as a ⋅ (b × c), gives the volume of the parallelepiped formed by the three vectors. If the volume is zero (meaning the vectors are coplanar), the scalar triple product equals zero.
7. Are all vectors in a plane automatically coplanar?
Yes, by definition, vectors lying in the same plane are coplanar.
8. Can coplanar vectors relate to forces in physics problems?
Yes, in physics, coplanar forces are forces whose lines of action lie in the same plane. Analyzing coplanar forces simplifies calculations of resultant forces and torques.
9. What are examples of coplanar vectors?
Examples include:
• Vectors representing forces acting on a rigid body in a plane.
• Vectors defining the sides of a triangle or polygon in a plane.
• Vectors used to represent points on the same plane in a 3D coordinate system.
10. Can vectors be coplanar if they are not concurrent?
Yes, coplanarity does not require vectors to share a common point (being concurrent). They only need to lie within the same plane.
11. Is coplanarity a property in higher-dimensional spaces?
Yes, the concept of coplanarity extends to higher dimensions. In n-dimensional space, coplanar vectors lie within a two-dimensional subspace (plane).
12. What is the geometrical significance of coplanar vectors?
Geometrically, coplanar vectors represent vectors that can be contained within a single flat surface, extending infinitely in two dimensions. Their scalar triple product being zero indicates that these vectors do not define a three-dimensional volume.
