**Coplanar Vectors** are the vectors that lie on the **same plane**. These vectors lie on the same plane and are parallel to the same surface.

The **coplanarity of three vectors** is defined when their **scalar product is zero**.

A plane is a two-dimensional figure extending to infinity in a three-dimensional space.

The coplanarity of three vectors is a condition where three lines lying on the same plane are called coplanar.

Index

**Conditions for Coplanar Vectors**

In a 3d space, if

- The triple scalar product is zero; then, those three vectors are coplanar.
- They are linearly independent; then, the three vectors are coplanar.
- No more than two vectors of ‘n’ are linearly independent; then all vectors are coplanar.

In the case of \(n\) vectors, if the **coefficient’s determinant is zero** or the coefficient’s matrix is a singular matrix, it is called a **non-trivial solution**.

And if the equation system has a **determinant of the coefficient as a non-zero**, yet the solutions are \(x=y=z=0\), then it is known as a **trivial solution**.

**Applications of ****Coplanar Vectors**

**Coplanar Vectors**

Coplanar vector is a **mathematical structure** and are widely used in the fields of **Mathematics, Physics, Engineering** Etc.

**Solved Examples**

**Question 1.** Are the vector \(x = { 1, 1, 1}, y = {1, 3, 1} \, \& \, z = {2, 2, 2}\) coplanar?

**Solution.** Calculating triple scalar product,

\(x \cdot [y\times z] = (1)\cdot[(3\cdot2) – (2\cdot1)] – (1)\cdot[(1\cdot2) – (2\cdot 1)] + (1)\cdot[(1\cdot 2) – (2 \cdot 3)]\).

\(x \cdot [y\times z] = 0\).

As the triple product is zero, x, y & z are coplanar.

**Question 2.** Are the vector \(x = { 2, 0, 3}, y = {2, 1, 1} \, \& \, z = {1, -1, 1}\) coplanar?

**Solution.** Calculating triple scalar product,

\(x \cdot [y\times z] = (2)\cdot[(1\cdot 1) – (-1 \cdot 1)] – (0)\cdot[(2 \cdot 1) – (1 \cdot 1)] + (3)\cdot[(2 \cdot 1) – (1 \cdot 1)]\).

\(x \cdot [y\times z] = 7\).

As the triple product is non-zero, x, y & z are not coplanar.

**FAQs**

**What are coplanar vectors?**Coplanar vectors are vectors what lye on the same plane. These vectors lie on the same plane and are parallel to the same surface. In a 3d-space, we can define this as a plane.

**Are all collinear vectors coplanar?**Collinear vectors are linearly independent. Three given vector are coplanar if they are linearly dependent (or) if their scalar triple product is zero.

**Whats is the meaning of trivial?**Solutions involving the number 0 are considered trivial. Nonzero solutions are considered nontrivial.

For example, the equation 3x + 7y = 0 has the trivial solution; x = 0, y = 0.

**What is a vector?**Vectors are used in science to describe anything that has both a direction and a magnitude.