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# NCERT Solutions for Class 12 Maths Chapter 10 – Vector Algebra

NCERT Solutions for Class 12 Maths Chapter 10 – Vector Algebra has been drafted by an expert Maths teacher on the basis of the latest CBSE Class 12 Maths Syllabus. Some of the significant concepts covered in this chapter of NCERT Solutions include an introduction to vector, types of vectors, components of a vector, the addition of vectors, subtraction of vectors, properties of vector addition, multiplication of a vector, section formula, the cross product of two vectors, dot product of two vectors, projection of a vector on a line, vector joining two points, and multiplication of a vector by a scalar. Our NCERT Solutions explicitly elaborate all concepts contained in this chapter with step by step solved examples.

CBSE NCERT Class 12 Maths Chapter 10 comprises 4 exercises and 1 miscellaneous exercise that covers all corners of the chapter. It gives you ample practise to master this chapter’s concepts and clear all your doubts. Our NCERT Solutions answer all questions of all exercises in an easy to understand and in a step by step manner. They have been specifically devised to help you strengthen your conceptual base and perform brilliantly in your Class 12 board exams. They are moduled to make learning more fun, interesting, stimulating, and an effective process.

## Introduction to NCERT CBSE Class 12th Maths Chapter 10

In NCERT Solutions for Class 12 Maths Chapter 10-Vector Algebra, you will study the d different aspects and concepts of vector algebra that are necessary to develop a strong foundation in this topic. Some of the topics covered in this chapter have been discussed below.

• Vector

In this section of CBSE NCERT Class 12 Maths Chapter 10, you will learn the definition of a vector and its various types. You will discover that a quantity that has magnitude, as well as direction, is called a vector. Some of the types of vectors you will learn here are Zero Vector, co-initial vectors, collinear vectors, equal vectors, and negative of a vector.

This chapter also teaches about various aspects related to vectors, such as the position of a vector, magnitude of a vector, physical representation of vectors, and how to find a vector when its position vectors of endpoint are given.

• Operations on vector

This section of the chapter teaches you in detail about various operations on vectors. Some of the operations learned in this chapter are as follows.

Here you will learn how two vectors can be added or subtracted. Here you will also learn the commutative law and the associative law.

The commutative law states that the order of addition of vectors does not matter.i.e a+b=b+a

The associative law states that the sum of any three vectors has nothing to do with which pair of the vectors is added in the beginning. i.e (a+b)+c= a+(b+c).

This section also tells you how you must know about reverse vectors to conduct a vector subtraction. It tells you that a reverse vector say (-a) is the opposite of a vector (a). It has a similar magnitude. However, it is pointed in the opposite direction. Hence we first find the reverse vector and then add them as usual.

Let’s say we want to find vector b-a

Then, b-a=b +(-a)

Vector Multiplication

Dot Product of two vectors

This section will teach you that the dot product of two vectors is the product of the cos of the angle between them and the magnitude of the two vectors. This section of Class 12 Maths Vector Algebra also specifies various dot product properties of the vector. Some of them have been listed below.

Dot Product Properties of Vector:

• Property 1: Dot product of two vectors is commutative i.e. a.b = b.a = ab cos θ.
• Property 2: If a.b = 0 , then it can be clearly seen that either b or a is zero or cos θ = 0 θ = π2. It suggests that either of the vectors is zero or they are perpendicular to each other.
• Property 3: Also we know that using the scalar product of vectors (pa).(qb)=(pb).(qa)=pq a.b
• Property 4: The dot product of a vector to itself is the magnitude squared of the vector, i.e., a.a=a.a  cos 0 = a2
• Property 5: The dot product follows the distributive law also i.e. a.(b+c) = a.b + a.c
• Property 6: In terms of orthogonal coordinates for mutually perpendicular vectors it is seen that i^.i^ = j^.j^= k^.k^ =1
• Property 7: In terms of unit vectors if a= a1i^+a2j^+a3k^ and b=b1i^+b2j^+b3k^ then
a.b=(a1i^+a2j^+a3k^).(b1i^+b2j^+b3k^)
a1b1+a2b2+a3b3 = ab cosθ

Cross product of two vectors

The mathematical value of the cross product of two vectors is represented with the following formula.

a×b = |a||b|sinθ n^, where a×b

where,

| a | is the magnitude of vector a.

| b | is the magnitude of vector b.

θ is the angle between two vectors a & b.

and n^ is a unit vector showing the direction of the multiplication of two vectors.

1. Parallelogram Law of vector addition: If two vectors a  & b are represented by adjacent sides of a parallelogram in magnitude and direction, then their sum a + b  is represented in magnitude and direction by the diagonal of the parallelogram through their common initial point.

### NCERT Class 12 Maths Chapter 10 Exercises

This chapter comprises 4 comprehensive exercises. Our NCERT Solutions for Class 12 Maths Chapter 10-Vector Algebra contain step by step solutions to all questions of the exercises. The exercises have been discussed below.

Exercise 10.1

This exercise comprises 5 questions. All five questions deal with the introduction of vectors and basic concepts of vectors and types of vectors. In the first question, you will have to find out the displacement of a vector from the given figure. The second and third questions ask you to classify given quantities into scalars and vectors. In Question 4, you will have to identify vectors and categorize them accordingly as per the given figure. Question 5 is a true-false that includes statements related to vector algebra.

Exercise 10.2

Exercise 10.2 consists of 19 questions. Some of the questions need you to calculate the magnitude of the vectors from the given equation, and some ask you to prove the equality of the vectors. Some of the questions need you to calculate the sum of vectors, and some need you to identify the direction of vectors. This exercise comprises both long answer questions and short answer questions.

Exercise 10.3

Exercise 10.3 consists of 18 questions. Some of the questions of this exercise need you to calculate the angles between two vectors with the given magnitude. The questions of this exercise deal with various operations of vectors and the section formula.

Exercise 10.4

Exercise 10.4 consists of 19 questions. This exercise deals with the method of determining perpendicular and parallel components of a vector, the various types of multiplication of vectors, and related theorems.

This chapter also comprises a  miscellaneous exercise that consists of 19 questions.

### Benefits of Class 12 Maths Chapter 10 Vector Algebra Solutions

Learning, revising, and practising from our comprehensive NCERT Solutions for Class 12 Maths Chapter 10-Vector Algebra has many upsides. Some of them have been given below.

• Our NCERT Solutions are well-structured, easy to understand, a hundred percent accurate, and of premium quality.
• They include all concepts of the chapter in an exhaustive yet to the point manner.
• They answer all questions from all exercises in a step by step manner to help you learn quickly and effectively.
• They cover a wide range of question types important from an exam point of view.
• Our smart study techniques and learning shortcuts help you learn quickly and score high marks in your Class 12 board exams and various competitive exams.
• Our NCERT Solutions have been designed to clear all your doubts and improve your retention rate.
• All questions have been solved using the best methods of solving the problems.
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