## Coplanar Vectors Introduction

Coplanar vectors are defined as vectors that are lying on the same in a three-dimensional plane. The vectors are parallel to the same plane. It is always easy to find any two random vectors in a plane, which are coplanar. Coplanarity of two lines lies in a three-dimensional space, which is represented in vector form. The coplanarity of three vectors is defined when their scalar product is zero.

### All about Coplanar Vectors

Coplanar Vectors is the concept that is taught in classes 11 and 12th respectively. The lesson belongs to a mathematical theory which is quite complex but has been explained in an easy and comprehensible way by the subject experts at Vedantu that will not only help the students to overcome their doubts but also improve their level of knowledge about the concept with ease.

The students can easily download the pdf available at Vedantu’s website along with other study material for no cost. The resources help the students to clear their concerts, learn new insights, understand the complexities, and much more to promote their overall growth.

For the candidates of class 12 it is discussed here that a vector has direction and a magnitude both however, the scalar has only magnitude. so, the Magnitude of a vector is denoted by |a| or a and it is a non-negative scalar.

### Equality of Vectors

Two vectors say for example- a and b are said to be equally written as a = b, if they have (i) same length (ii) the same or parallel support, and (iii) the same sense.

### Types of Vectors

There are the following types of Vectors in the chapter Vectors:

(i) Zero or Null Vector- It is a vector that has initial and terminal points as coincident, and is called zero or a null vector. Also, it is denoted by 0.

(ii) Unit Vector- this is a type of Vector whose magnitude is unity and is called a unit vector. This is denoted by nˆ

(iii) Free Vectors- for these Vectors if the initial point of a vector is not specified, then they are said to be a free vector.

(iv) Negative of a Vector- A vector which has the same magnitude as that of a given vector say ‘a’ and the direction opposite to that of ‘a’ is known as the negative of ‘a a’ and is denoted by —a.

(v) Like and Unlike Vectors- the Vectors are said to be like when they have the same direction and unlike when they have opposite direction.

(vi) Collinear or Parallel Vectors- these Vectors have the same or parallel supports and are called collinear vectors.

(vii) Coinitial Vectors- The Vectors having the same initial point are known as coinitial vectors.

(viii) Coterminous Vectors- these are the Vectors having the same terminal point and are called coterminous vectors.

(ix) Localized Vectors- A vector that is drawn parallel to a given vector through a specified point in space is called a localized vector.

(x) Coplanar Vectors- A system of vectors are said to be coplanar if their supports are parallel to the same plane. However, otherwise, they are known as non-coplanar vectors.

(xi) Reciprocal of a Vector- A vector that has the same direction as that of a given vector but the magnitude that is equal to the reciprocal of the given vector is called the reciprocal of ‘a’.

i.e., if |a| = a, then |a-1| = 1 / a.

### Coplanarity in Theory

Coplanar lines are a very common topic in three-dimensional geometry. In mathematical theory, the coplanarity of three vectors is called a condition where three lines lying on the same plane are referred to as coplanar.

A plane is a two-dimensional figure going into infinity in the three-dimensional space, while we have used the straight lines as vector equations.

### Conditions for Coplanar Vectors/ Properties of Coplanar Vectors

If three vectors are coplanar then their scalar product is zero, and if these vectors are existing in a 3d- space.

The three vectors are also coplanar if the vectors are in 3d and are linearly independent.

If more than two vectors are linearly independent; then all the vectors are coplanar.

So, the condition for vectors to be coplanar is that their scalar product should be 0, and they should exist on 3d; then these vectors are coplanar.

The equation system that has the determinant of the coefficient as zero is called a non-trivial solution. The equation system that has a determinant of the coefficient matrix as non zero, but the solutions are x=y=z=0 is called a trivial solution.

### What are Linearly Independent Vectors?

The vectors, v1,……vn, are linearly independent when the non-trivial combination of the vectors is a zero vector.

a1v1 + … + an vn = 0 where the coefficients a1= 0 ….. an=0.

### What are Linearly Dependent Vectors?

The vectors, v1,……vn, are linearly dependent when there is at least one non-trivial combination of these vectors, which is equal to zero vector.

### How do you Know if Two Vectors are Coplanar?

If the scalar triple product of any three vectors is 0, then they are called coplanar. The vectors are coplanar if any three vectors are linearly dependent, and if among them not more than two vectors are linearly independent.

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1. Solve for the Value of x When Point A(3,2,1) B (4,x,5) C (4,2,-2) and D (6,5,-1) are Coplanar.

Given, points are: A(3, 2, 2), B(2, x, 5), C (4, 2, -2) and D (6, 5, -1)

Now, AB = position vector of B - position vector of A

=> AB = (4i + xj + 5k) - (3i + 2j + 2k) = i + (x - 2)j + 3k

AC = position vector of C - position vector of A

=> AC = (4i + 2j - 2k) - (3i + 2j + 2k) = i + 0j -4k

Now, AD = position vector of D - position vector of A

=> AD = (6i + 5j - k) - (3i + 2j + 2k) = 3i + 3j - k

As A, B, C and D are coplanar, then

|AB AC AD| = 0

=> |1 x-2 3|

|1 0 -4| = 0

|3 3 -1|

=> 1(0 +12) - (x - 2)*(-1 + 12) + 3(3 - 0) = 0

=> 12 - 11(x - 2) + 3 * 3 = 0

=> 9 - 11x + 22 + 9 = 0

=> 39 - 7x = 0

=> 7x = 39

=> x = 39/7

2. Find the Scalar Triple Product of Vectors i + 2j + 3k, - i - 2j + k and i + k

|1 2 3|

|-1 -2 1| = 0

|1 0 1|.

Scalar triple product = 1(-2-1)- 2 (-1-1)+ 3(0+2)

=1(-3)- 2 (-2)+ 3(2)=-3+4+6 =7

3. Prove that [a+b,b+c,c+a]=2[a,b,c]

LHS=(a+b).{(b+c)X(c+a)}

=(a+b).{bxc+bxa+cxc+cxa}

=(a+b).{bxc-axb+cxa}

=a.(bxc)-a.(axb)+a.(cxa)+b(bxc)-b(axb)+b.(cxa)

=a.(bxc)-0+0+0-0+b.(cxa)

=a.(bxc)+a(bxc) = 2a.(bxc)=2[a,b,c]

RHS = 2[a,b,c]

4. Search the Value of y if the Vectors i+j+2k,yi-j+k and 3i-2j-k are Coplanar.

Given that vectors are coplanar

|1 1 2|

|y -1 1|

|3 -2 - 1|.

= 1 (1 + 2 ) - 1 ( - y - 3 ) + 2 ( - 2 y + 3 ) = 0

3 + y + 3 - 4y + 6 = 0

-3y + 12= 0

y = 4

5. Define Coplanar Vectors?

Coplanar vectors are defined as vectors that exist on the same in a three-dimensional plane. These vectors are always parallel to the plane. Also, it is easy to find any two random vectors in a single plane, which are coplanar. The Coplanarity of the two lines lies in a three-dimensional space, which is represented in vector form. For more information about the Coplanar vectors, the students can refer to Vedantu’s website. There they will find several other study materials for free in PDF format.