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Coplanarity of Vectors

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Last updated date: 13th Jul 2024
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What are Vectors and What is Coplanarity of Vectors?

Geometrical entities with magnitude and direction are known as vectors. A vector is represented as a line with an arrow pointing in the direction of the vector, and the length of the line denotes the vector's magnitude. As a result, vectors are represented by arrows and have two points: an initial point and a terminal point. Over 200 years, the concept of vectors has changed. Physical quantities such as displacement, velocity, and acceleration are represented by vectors.

In addition, the application of vectors began in the late nineteenth century with the development of electromagnetic induction. For a better understanding, we will look at the definition of vectors, as well as their properties, formulas, and operations, as well as solved instances.

A vector is a quantity or phenomenon with magnitude and direction that are independent of one another. The phrase also refers to a quantity's mathematical or geometrical representation.

Velocity, momentum, force, electromagnetic fields, and weight are all examples of vectors in nature. (The force produced by gravity's acceleration acting on a mass is known as weight.) A Scalar is a quantity or phenomenon that has merely magnitude and no particular direction. Speed, mass, electrical resistance, and hard-drive storage capacity are examples of scalars.

Vectors can be shown in two or three dimensions graphically. The length of a line segment is used to represent magnitude. The orientation of the line segment, as well as an arrow at one end, indicate direction. Three vectors in two-dimensional rectangular coordinates (the Cartesian plane) and their polar analogues are shown in the diagram.

Linearly Dependent and Independent Vectors: 

A linear combination of vectors a1, ….., 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 Dependent Vectors: 

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

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 applications of vectors:

  • The direction in which force is applied to move the object is found using vectors.

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

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

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

  • They are found everywhere in general relativity.

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

  • Vector has its application on Quantum Mechanics

  • Vectors are used in various oscillators

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

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. 

Types of Vectors

  •  Zero Vector.

  •  Unit Vector.

  •  Position Vector.

  •  Co-initial Vector.

  •  Like Vector. 

  •  Unlike Vectors.

  •  Coplanar Vector.

  •  Collinear Vector.

Let’s understand some types of vectors

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.

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.

Unit Vector: 

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

Collinear Vector: 

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

Coplanar Vectors: 

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

What are Coplanar Vectors? 

Coplanar vectors can be defined as the type of vectors that lie on the same plane and these are also parallel to the same surface. Three vectors are said to be coplanar if their scalar product is zero. In three-dimensional space, a plane is a two-dimensional figure that extends to infinity. Three vectors' coplanarity is a condition in which three lines on the same plane are coplanar.

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 math. 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 math. 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

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:

  • Two vectors are always coplanar

  • Collinear vectors are linearly independent

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

Coplanar Vectors' Requirements

If the triple scalar product in a three-dimensional space is zero, the three vectors are coplanar. The three vectors are coplanar because they are linearly independent. Only two 'n' vectors are linearly independent; after that, all vectors are coplanar. A non-trivial solution is one in which the coefficient's determinant is zero or the coefficient's matrix is singular in the case of n vectors. And if the determinant of the coefficient is non-zero but the solutions are x = y = z = 0, the equation system is said to be trivial.

Uses of Coplanar Vectors 

Coplanar Vectors have a variety of applications. Coplanar vectors are a type of mathematical structure that is commonly used in domains such as mathematics, Physics, and engineering.

FAQs on Coplanarity of Vectors

1. What are the various applications of vectors? 

There are a lot of applications of vectors. These can be used in the field of engineering, geometry, Physics and some other fields as well. These vectors can be used to find the direction of the application of force. You can find the motion of that body which is confined to a plane with the help of vectors and can also use them to understand the effect of gravity on all objects. 

2. What is the difference between linearly dependent and independent vectors? 

Two vectors are said to be linearly dependent when at least one of those two vectors is a multiple of the other or if there is a combination (non-trivial) of these two vectors which is equal to zero vector. Similarly two vectors are said to be linearly independent if neither of these two vectors is a multiple of the other or in other words if these two vectors have no combination (non-trivial). 

3. What are the different types of vectors? 

There are different types of vectors and these types are zero vectors, unit vectors, collinear vectors, coplanar vectors, etc. Zero vectors are those vectors which have the same starting and endpoints, unit vectors are those vectors that have a  magnitude equal to 1, equal vectors are those vectors that have the same magnitude as well as the same direction. Vectors that lie on the same line are said to be collinear vectors. Similarly, vectors that lie in the same plane are called coplanar vectors.

4. How to Prove Vectors are Coplanar?

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

  • 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.

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

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

5. How to Find the Coplanarity of Two Vectors?

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 the 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.