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# NCERT Solutions for Class 12 Maths Chapter 10 Exercise 10.3 Vector Algebra - PDF Download

NCERT solutions for class 12 maths chapter 10 exercise 10.3 Vector Algebra is based on the topic product of two vectors such as the scalar or dot product of two vectors and projection of a vector on a line. As you already have learnt that function can be multiplied in two ways namely, multiplication of two functions pointwise and composition of two functions. Similarly here, you will gain a deep knowledge of multiplication of two vectors in two ways. Subject matter experts of eSaral have prepared all the concepts in simple language for students to score good marks in exams.

Class 12 maths chapter 10 exercise 10.3 NCERT solutions have a total of 18 questions that cover all of the topics mentioned above to develop a strong foundation of concepts in different levels of questions. By practicing these questions provided by expert teachers of eSaral will help you prepare for exams and make your learning easy.

Ex 10.3 class 12 maths chapter 10 solutions are also made available here in PDF format. You can download these PDFs for free from the official website of eSaral and practice all the questions of ex 10.3 without being confused. We have provided below the link to download the NCERT solution PDF for ex 10.3 class 12 maths ch 10.

## Topics Covered in Exercise 10.3 Class 12 Mathematics Questions

NCERT solution for ex 10.3 class 12 maths has some essential topics covered such as product of two vectors, scalar (or dot) product of two vectors and projection of a vector on a line. We have explained these topics in detail for your convenience.

 1 Product of Two Vectors Scalar (or dot) product of two vectors Projection of a vector on a line
1. Product of Two Vectors

Multiplication of two vectors is defined in two ways, namely, scalar (or dot) product where the result is a scalar, and vector (or cross) product where the result is a vector. Based upon these two types of products for vectors, they have found various applications in geometry, mechanics and engineering. Here, you will get to know these two types of products.

• Scalar (or dot) product of two vectors

The scalar product of two given vectors $\vec{a}$ and $\vec{b}$ having angle θ between them is defined as

$\vec{a} \cdot \vec{b}=|\vec{a}||\vec{b}| \cos \theta$

Also, when $\vec{a} \cdot \vec{b}$ is given, the angle ‘θ’ between the vectors $\vec{a}$ and $\vec{b}$ may be determined by

$\cos \theta=\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$

Two important properties of scalar product

Property 1 (Distributivity of scalar product over addition) Let $\vec{a}, \vec{b}$ and $\vec{c}$ be any three vectors, then

$\vec{a} \cdot(\vec{b}+\vec{c})=\vec{a} \cdot \vec{b}+\vec{a} \cdot \vec{c}$

Property 2 Let $\vec{a}$ and $\vec{b}$ be any two vectors, and l be any scalar. Then

$(\lambda \vec{a}) \cdot \vec{b}=(\lambda \vec{a}) \cdot \vec{b}=\lambda(\vec{a} \cdot \vec{b})=\vec{a} \cdot(\lambda \vec{b})$

• Projection of a vector on a line

A vector $\overrightarrow{\mathrm{AB}}$ makes an angle θ with a given directed line l (say), in the anticlockwise direction. Then the projection of $\overrightarrow{\mathrm{AB}}$ on l is a vector $\vec{p}$ (say) with magnitude $|\overrightarrow{\mathrm{AB}}||\cos \theta|$ , and the direction of $\vec{p}$ being the same (or opposite) to that of the line l, depending upon whether cosθ is positive or negative. The vector $\vec{p}$ is called the projection vector, and its magnitude $|\vec{p}|$ is simply called as the projection of the vector $\overrightarrow{\mathrm{AB}}$ on the directed line l.

## Tips for Solving Exercise 10.3 Class 12 Chapter 10 Differential Equations

Ex 10.3 class 12 maths chapter 10 can be solved by following the tips provided by experts of eSaral.

1. Ex 10.3 class 12 maths has two main properties associated with the scalar (or dot) product of two vectors that you need to be well-versed to solve questions of exercise 10.3.

2. You should also go through the projection of a vector on a line to understand the concept properly for solving questions in this exercise.

3. Additionally, you must practice all the examples given before the exe 10.3 for better comprehension of concepts.

## Importance of Solving Ex 10.3 Class 12 Maths Chapter 10 Differential Equations

You will gain numerous benefits of solving questions in ex 10.3 class 12 maths chapter 10 Differential Equations. Our subject experts have provided some of them for you which can be checked below.

1. Every topic and concept is deeply explained by experienced teachers of eSaral in NCERT solutions class 12 maths chapter 10 ex 10.3. Therefore, you will feel confident in the examination while solving questions related to this chapter.

2. All the questions of ex 10.3 class 12 maths are elaborated and solved in a step by step manner that will help you to understand the concept behind each question.

3. NCERT solutions for ex 10.3 class 12 maths chapter 10 are also available in free PDF which you can download anytime and solve all the questions error-free.

4. By solving questions in ex 10.3 class 12 maths, you will be able to improve your mathematical skills that will be beneficial in board exams.

Question 1. What are the two main properties of scalar product?

Answer 1. Two main properties of scalar product are as follows.

1. Distributivity of scalar product over addition

$\vec{a} \cdot(\vec{b}+\vec{c})=\vec{a} \cdot \vec{b}+\vec{a} \cdot \vec{c}$

1. $(\lambda \vec{a}) \cdot \vec{b}=(\lambda \vec{a}) \cdot \vec{b}=\lambda(\vec{a} \cdot \vec{b})=\vec{a} \cdot(\lambda \vec{b})$

Question 2. What are the key topics explained in exercise 10.3 class 12 maths chapter 10?

Answer 2.  The key topics of exercise 10.3 class 12 maths chapter 10 are as follows.

1. Scalar (or dot) product of two vectors

2. Projection of a vector on a line