Coplanarity of Vectors

What are Coplanar Vectors?

The vectors which are parallel to the same plane or lie on the same plane are said to be coplanar. It is always possible to find a plane parallel to two random vectors. Any two random vectors in a plane are coplanar. 

A vector is an object in the geometry which has magnitude and direction both. Magnitude is the size of the vector. It plays an important role in physics, engineering as well as maths. Vectors are equal if their magnitude and direction are equal. They are said to be equal in accomplishing the statement.

Coplanar Vectors Definition

Vector is simply defined as an object which contains both magnitude and direction. It describes the movement of an object from one direction to another. The starting point of a vector is called the tail and the ending point is called the head.

It is a mathematical structure and has many applications in the field of physics, engineering, and maths. The location of points on the coordinate plane is represented by ( x,y ). Usage of the vector is very useful to simplify the process of three-dimensional geometry. 

Coplanar vector: Three or more vectors lying in the same plane are known as coplanar vectors

Types and Examples of Vector

Velocity, acceleration, force, rise, or decrease in temperature. All these quantities have magnitude and direction both. Speed being the unit has only magnitude and no direction. This is the basic difference between speed and velocity. 

Let’s understand some types of vectors

  1. Zero Vector: 

The vector whose starting point and endpoint coincide is known as the zero vector. It is denoted by 0 and has no magnitude.

  1. Equal Vector: 

If two or more vectors are on the same parallel or line. These are said to be equal vectors. They should also follow in the same direction as well.

  1. Unit Vector: 

Those vectors whose length is equal to one is said to be a unit vector.

  1. Collinear Vector: 

Vectors that are parallel to one line or are lying on the same line are known as collinear vectors.

  1. Coplanar Vectors: 

Those vectors which are parallel to the same plane are denoted as coplanar vectors.

Condition for Coplanarity of Vectors

Following are the condition when vectors are termed as coplanar

  1. If the scalar triple product of any three vectors is zero, then they are considered as coplanar

  2. If any three vectors are linearly dependant, they are coplanar

  3. Vectors are considered coplanar if amongst them no more than two vectors are linearly independent vectors.

Linearly Dependent and Independent Vectors: 

A linear combination of vectors a1, ….., a with coefficients is a vector. A linear combination x1a1 is called trivial if all the coefficients x1… are zero and is called non-trivial if at least one of them is not zero

Linearly Independent Vectors: 

The vectors a1… an are called linearly independent if there is no non-trivial combination. The vector is linearly independent if x1a1 + …. + xnan = 0, if x1 = 0, … xn = 0.

Linearly Dependant Vectors: 

The vectors a1, …, and are linearly dependent if there is a non-trivial combination of these vectors is equal to zero vector.

Examples of Coplanar Vectors 

1. Check whether the following vectors are coplanar or not a= ( 1,2,3 ), b= ( 1,1,1 ), c= ( 1,2,1 ).

Ans: Here, vectors are not coplanar as their scalar triple product is not zero.

Points to look for on Coplanar vectors:

  1. Two vectors are always coplanar

  2. Collinear vectors are linearly independent

  3. Three given vectors are coplanar if they are linearly dependent or if their scalar triple product is zero.

Application of Vectors:

As discussed above, vectors are used in the field of physics, engineering, and geometry. It has applications in real life too. Following are the points which will discuss some real-life application of vectors:

  1. The direction in which force is applied to make movement in the object is found using vectors.

  2. It is used to understand how gravity uses the force of attraction on an object.

  3. The  motion of a body confined to the plane is obtained using vectors

  4. It is used in wave propagation, sound propagation, vibration propagation, etc.

  5. They are found everywhere in general relativity.

  6. The velocity in the pipe is determined in terms of the vector field.

  7. Vector has its own application on Quantum Mechanics

  8. Vectors are used in various oscillators

  9. Vectors help in defining the force applied to a body in the three dimensions.

FAQs (Frequently Asked Questions)

Q1. How to Prove Vectors are Coplanar?

Ans: There are the following conditions to prove if the vector is coplanar or not. These conditions are as follows:

  1. If there are three vectors in a three-dimensional space and the scalar triple product is zero, these three vectors are said to be coplanar.

  2. If there are three vectors in a three-dimensional space that are linearly independent, these three vectors are coplanar.

  3. In the case of n vectors, if no more than two vectors are linearly independent. Then all three vectors are coplanar.

Q2. How to Find the Coplanarity of Two Vectors?

Ans: Two or more vectors are coplanar if they satisfy linearly dependent conditions. Their components are proportional and the rank denoted is 2. 

Two or more points are coplanar if the vectors determined by them are also coplanar.

For example: Determine if the points A= (1,2,3) ,B= (4,7,8) ,C= (3,5,5) ,D= (-1,-2,-3) ,E= (2,2,2) are coplanar.

The points A,B,C,D,E are coplanar if rank AB, AC, AD, AE = 2

AB = ( 4-1 , 7-2 , 8-3 ) = ( 3,5,5 )

AC = ( 3-1 , 5-2 , 5-3 ) = ( 2,3,2 )

AD = ( -1-1 , -2-2 , -3-3 ) = ( -2,-4,-6 )

AE = ( 2-1 , 2-2 , 2-3 ) = ( 1,0,-1 )

Rank ( AB, AC, AD, AE ) = 3

The points A, B, C, D, and E are not coplanar as it does not have pre-mentioned rank, that is  2.When two or more vectors are coplanar, their components are proportional and their rank is 2. Two or more points are coplanar if the vectors determined by them are also coplanar.