Coplanar Vectors

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.

Conditions for Coplanar Vectors

In a 3d space, if

1. The triple scalar product is zero; then, those three vectors are coplanar.
2. They are linearly independent; then, the three vectors are coplanar.
3. 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 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